I would like to argue about .999

[ramsey2879]

No, to get the limit of A/n as n goes to infinity, you need to know the Archimedean principle.

To compute the limit of A/n as n goes to infinity instead by comparing to A/[infinity], you need a lot more information. One set of information would be

• A number system containing an element called [infinity] along with all real numbers
• Knowledge that A/[infinity] = 0 if A is finite
• Knowledge that division in this new number system gives the same results as division in the real numbers, when both numbers are real
• Knowledge that division is continuous in this new number system (at least, at (A, [infinity]))
• Knowledge that the limit of n as n goes to infinity converges to [infinity]
• Knowledge that limits computed in this new number system agree with limits computed in the real numbers when it would make sense.

(For the record, my thought processes probably would compute the limit by invoking continuity of division in the projective real numbers before any other approach)

I supposed that calculus back in high school in the 1960's may have treated infinity differently from how the treat it today. I believe that all we had to do was simply recognized that infinity was like a much much bigger number than A to deduce that A/[infinity] was 0. PS I don't understand Micromass's math notation as I never had much math beyond High School.

 

[JaredJames]

I believe that all we had to do was simply recognized that infinity was like a much much bigger number than A to deduce that A/[infinity] was 0.

Not saying it's right or wrong, but that's also how I was taught to treat infinity in this situation.

 

[statdad]

" infinity was like a much much bigger number than A"

The problem is that when you are working with the real number system infinity is not a number, so performing calculations with it makes no mathematical sense.

 

[Mark44]

I supposed that calculus back in high school in the 1960's may have treated infinity differently from how the treat it today. I believe that all we had to do was simply recognized that infinity was like a much much bigger number than A to deduce that A/[infinity] was 0.

I was in high school in the 60s also, but I don't recall that we were told to treat infinity as just a big number. No, I don't believe that infinity was presented any differently back then as compared to now.

" infinity was like a much much bigger number than A"

The problem is that when you are working with the real number system infinity is not a number, so performing calculations with it makes no mathematical sense.

Right, and this was my point to ramsey2879. An expression such as n/[infinity] explicitly uses infinity in the division, which isn't valid.

 

[JaredJames]

I was in high school in the 60s also, but I don't recall that we were told to treat infinity as just a big number. No, I don't believe that infinity was presented any differently back then as compared to no.

I was in school with this stuff 5 years ago.

I was always told that if you see something over infinity it's just like having an extremely big number so you just assume 0.

Again, not arguing this either way, just pointing out the way it was taught.

 

[Raphie]

So... uh... where do "fluxions" figure into this recent discussion?

Is there a place in mathematics for what one might term "virtual" infinity or "virtual" zero?

- RF

Definition: "Virtual Infinity": The largest number one can possibly imagine at a given point in time, plus 1, at a point in time just subsequent to the point in time one initially imagined... ( e.g. Skewe's Number + 1)

 

[Hurkyl]

So... uh... where do "fluxions" figure into this recent discussion?

They don't.

Is there a place in mathematics for what one might term "virtual" infinity or "virtual" zero?

- RF

Definition: "Virtual Infinity": The largest number one can possibly imagine at a given point in time, plus 1, at a point in time just subsequent to the point in time one initially imagined... ( e.g. Skewe's Number + 1)

That's not a very good definition. :tongue:

For the record, the most basic technique of calculus/analysis is the idea of "close enough" or "big enough". You don't have to invoke some mythical number larger than you can imagine; you just need to invoke a number that is big enough for the purpose at hand.

(To guard against misinterpretation, I will point out that infinity is not mythical -- at least, the mathematical notions of infinity are not. Many laypeople seem to have some mythical notion of it, though. )


Anyways, I know the basic idea you seem to be thinking. Some people like to attach a philosophical interpretation to non-standard analysis where, for example, the "standard integers" are the ones accessible to mathematicians and all other integers are simply too big or complicated for mathematicians to access directly. Wikipedia's page on internal set theory describes this viewpoint.

And for the record, in the non-standard model, while there are only finitely^* many standard integers, there does not exist a number that says how many there are. In particular, there isn't a largest one.

Also, for the record, non-standard analysis doesn't need this philosophical interpretation. It is entirely optional. In fact, I have only ever seen it mentioned when the speaker wants to model the idea of "what is accessible to mathematicians" and sometimes in the context of internal set theory.

*: by the non-standard 'measure' of such things. There are infinitely many standard integers by the standard 'measure', of course.

 

[Raphie]

That's not a very good definition. :tongue:

In a strict mathematical sense, I would concur, Hurkyl. Just trying to stake out some manner of middle ground here.

Too bad mathematical "language" does not allow for that middle ground between .99999 and .99999... Where is the possibility of the (sliding scale...) partial sum contained within that black/white notational dichotomy?

See post #22, by the way, for confirmation that we agree in principle on the basic question of this thread.

 

[Hurkyl]

In a strict mathematical sense, I would concur, Hurkyl. Just trying to stake out some manner of middle ground here.

Too bad mathmatical "language" does not allow for that middle ground between .99999 and .99999... Where is the possibility of the partial sum contained within that black/white notational dichotomy?

See post #22, by the way, for confirmation that we agree in principle on the basic question of this thread.

0.999999 is in the middle ground. So is 0.9999999 if the former didn't have enough 9's for you.

There is no middle ground between 0.999... and the set of all partial sums, though. This is a rather important geometric property of the number line.

Also, there is the sequence of numbers {1 - 10ⁿ} for those people who need to feel the need to consider a sequence of increasingly good approximations to 0.999...

And, of course, there is non-standard analysis for people who want a hyperfinite number of 9's, but still have a number that is only infinitessimally different from 0.999...
(hyperfinite for the non-standard measure of finiteness)


There is mathematical language for all sorts of ideas, and if there's not, it could be invented. The only true obstacle is when someone insists on grounding their reasoning firmly in the realm of vagueness and imprecision.

 

[Char. Limit]

0.999999 is in the middle ground. So is 0.9999999 if the former didn't have enough 9's for you.

There is no middle ground between 0.999... and the set of all partial sums, though. This is a rather important geometric property of the number line.

Also, there is the sequence of numbers {1 - 10ⁿ} for those people who need to feel the need to consider a sequence of increasingly good approximations to 0.999...

And, of course, there is non-standard analysis for people who want a hyperfinite number of 9's, but still have a number that is only infinitessimally different from 0.999...
(hyperfinite for the non-standard measure of finiteness)


There is mathematical language for all sorts of ideas, and if there's not, it could be invented. The only true obstacle is when someone insists on grounding their reasoning firmly in the realm of vagueness and imprecision.

I'm going to guess that that sequence is supposed to read {1 - 10^-n}.

 

[TylerH]

actually, there is a decimal expansion between .999... and 1. it's .9999... (the extra 9 and . signifying that it is still expanding and that there will always be a 9 between the two)

The number of 9s and the number of .s is arbitrary. They just mean to continue in the "obvious way." (Hence the ambiguity Fredrick keeps talking about.) The "obvious way," in this case, being a repeating decimal. In other words, ".999... = .9999... ." Although, it is commonly accepted that three is the correct number of .s for an "ellipsis," which is what ... is. The concept is also flawed. You are assuming ∞ + 1 > ∞ when, in fact, ∞ + c = ∞, where c is a constant. But ∞ isn't a real number, so one shouldn't use it as I just did. I was just illustrating my point. In other words, one more 9 than an infinite number of 9s is the same number of 9s as an infinite number of 9s.

how can the two meet if .999... is expanding onward forever? it would have to stop expanding to reach 1.

They meet only because .999... repeats forever. Numbers, or "expressions," don't "expand." They have a set value. You can define lots of ways to get that set value, the easiest is the infinite sum everyone keeps using. They don't just come up with it arbitrarily, it's origin comes from the very definition of the real numbers. See Wikipedia: http://en.wikipedia.org/wiki/Decimal_representation, in this case, we would define a=.9 for all i. Then we take the infinite sum, to get the same proof we've already seen 100 times, just with a little background behind why it's valid, this time. Does that help?

 

[disregardthat]

And, of course, there is non-standard analysis for people who want a hyperfinite number of 9's, but still have a number that is only infinitessimally different from 0.999...
(hyperfinite for the non-standard measure of finiteness)

Is 0.9999 = 1 in the non-standard reals? Is the non-standard reals even complete? I should think so by the transfer-principle, but it is clear that 0.99999... cannot converge since every term is less that 1-e for some infinitesimal e. If so, how does 0.999 even make sense in the non-standard reals? Maybe the transfer-principle doesn't apply to completeness.

EDIT: what does an hyperfinite number of 9's mean?

 

[Raphie]

There is no middle ground between 0.999... and the set of all partial sums, though. This is a rather important geometric property of the number line.

In a philosophical vein... That I recognize the validity of this statement (i.e "no middle ground") is what has had me thinking much of late about Durkheim's distinction between the sacred and the profane.

 

[Hurkyl]

If so, how does 0.999 even make sense in the non-standard reals? Maybe the transfer-principle doesn't apply to completeness.

When you transfer, you have to transfer everything; you can't pick and choose.

In the standard model, a decimal numeral has its places indexed by integers. When you transfer that notion to the non-standard model, the corresponding notion of a "hyperdecimal numeral" has its places indexed by hyperintegers.

The partial sums of the non-standard infinite summation

$$\sum_{n=1}^{+\infty} 9 \cdot 10^{-n}$$​

that would define the hyperreal value of the hyperdecimal numeral 0.999...
(don't forget n ranges over hyperintegers!) are well-defined (since everything appearing is internal), and it's easy to check that they are an increasing (internal) bounded sequence, that they satisfy the transfer of the Cauchy criterion for convergence of a sequence, and so forth.

Of course, it's easier to compute this sum by just recognizing that it's the transfer of a standard sum that converges to 1.

EDIT: what does an hyperfinite number of 9's mean?

It means that there is a hyperinteger H bigger than zero, and the n-th digit of the hyperdecimal numeral in question is:

• 9, if H <= n < 0
• 0 otherwise

(the 0-th place is the one's place, the 1-th place is the ten's place, the (-1)-th place is the tenth's place, etc)

 

[disregardthat]

Thanks. Does this mean that the corresponding thing to a countable sequence in the standard reals is a sequence indexed by hyper-integers?

9, if H <= n < 0
0 otherwise

And does this have a corresponding non-standard real number (I should guess so as it is a cauchy-sequence indexed by hyperintegers)? Does all non-standard real numbers have a corresponding digit representation?

Still, shouldn't every subset of the non-standard reals bounded below have a largest lower bound by the transfer principle? If we consider the subset of real numbers larger than 0 as a subset of the non-standard reals, this would have no largest lower bound in the non-standard reals. Am I not using the transfer-principle correctly here?

EDIT: Maybe it is because the subset of real numbers cannot properly defined by the transferred axioms of the reals to the nonstandard reals in order for the transfer principle to work? EDIT again: This got me wondering about the "standard part" function. If the reals is not a "properly defined" subset of the non-standards, the standard part couldn't be "properly defined" either.

 

[Hurkyl]

Thanks. Does this mean that the corresponding thing to a countable sequence in the standard reals is a sequence indexed by hyper-integers?

Yep. Well, two technicalities.

The first one is harmless -- in standard analysis, there are lots of countable ordinals (or even more general "order types") that could be used to index seqences. I'm assuming you didn't mean to include these more general sorts of things.

The second is more important -- the sequence has to be "internal".

And does this have a corresponding non-standard real number (I should guess so as it is a cauchy-sequence indexed by hyperintegers)? Does all non-standard real numbers have a corresponding digit representation?

Yep! It's the transfer of the standard theorem:

Every real number is equal to the infinite sum that computes the value of some decimal number​

which becomes

Every hyperreal number is equal to the infinite sum that computes the value of some hyperdecimal number​

Still, shouldn't every subset of the non-standard reals bounded below have a largest lower bound by the transfer principle?

Every internal set.

Am I not using the transfer-principle correctly here?

The internal / external distinction is probably the most important one to understand to avoid making mistakes.

(I think your edit is touching upon the ideas I write below down to the horizontal line)

In the non-standard model, we have the set of hyperintegers, the set of hyperreals, and so forth. And to these we can apply the tools set theory, calculus, analysis, number theory, algebra, or whatever. The transfer principle says the standard and non-standard models have exactly the same theorems.

The power of non-standard analysis comes because we can also view the hyperreals as a standard set. Even better, we can view the standard real numbers as a subalgebra of the non-standard ones!

But that's where the danger comes too -- viewed this way, the standard model has a lot more things in it than the non-standard model does. Doing set theory in the standard model let's us construct a lot of sets that the non-standard model doesn't have. (similarly for sequences, functions, et cetera)

(of course, this all transfers. The non-standard model has hyperhyperreals and a "hypernonstandard model" built on top of them...)
___________________________________________________

Generally speaking, there is a quick way to tell if an object is internal (and thus usable in the non-standard model) or not -- if it makes any reference whatsoever to the standard model other than transferring something, it's probably external.

So the standard part function is, in fact, external. The set of positive standard real numbers (viewed as a subset of the hyperreals) is also an external set. Because it is not internal, it doesn't contradict the LUB property that every internal nonempty bounded subset of the hyperreals has a greatest lower bound.

 

[disregardthat]

Thank you, these are really good explanations. I should get hold of a book on model theory, the transfer principle seems really powerful.

 

[Hurkyl]

Have you tried Keisler's calculus book? I've skimmed through part of it and found it useful. (but then, I knew some non-standard analysis already before doing so)

 

[disregardthat]

Have you tried Keisler's calculus book? I've skimmed through part of it and found it useful. (but then, I knew some non-standard analysis already before doing so)

I haven't, but this is excellent. Thanks for the reference, this will come in handy. Do you perhaps know of a good reference which treats the transfer principle as well?

 

[SewerRat]

Here's a proof that .999...=1.

.999... can be written as the infinite sum as follows:

<SNIP>

---

Hello Char - I found that little proof fascinating. Might that concept also extend to natural numbers like Pi, e, etc. ? Although they are not recurring as such, they are nevertheless convergent, getting ever closer to an asymptote value without ever reaching it (intuitively). If a proof can be established, it would yield actual values for said numbers, albeit with a huge number of decimal places but a finite number nonetheless.

 

[gb7nash]

---

Hello Char - I found that little proof fascinating. Might that concept also extend to natural numbers like Pi, e, etc. ? Although they are not recurring as such, they are nevertheless convergent, getting ever closer to an asymptote value without ever reaching it (intuitively). If a proof can be established, it would yield actual values for said numbers, albeit with a huge number of decimal places but a finite number nonetheless.

(Do you mean irrational numbers? pi and e aren't natural numbers :tongue:) The problem with pi is that you can't really establish a geometric series since there's no specific formula for calculating a certain digit of pi.

 

[Robert1986]

---

Hello Char - I found that little proof fascinating. Might that concept also extend to natural numbers like Pi, e, etc. ? Although they are not recurring as such, they are nevertheless convergent, getting ever closer to an asymptote value without ever reaching it (intuitively). If a proof can be established, it would yield actual values for said numbers, albeit with a huge number of decimal places but a finite number nonetheless.

To answer your question, the method used by char limit cannot be used to give a representation of $\pi$ with a finite number of decimal places, same for $e$. Both of these numbers are irrational, so there is no finite decimal expansion of them.

However, we can write both of them as the sum of an infinite number of terms, as Char Limit did. I will show you below, but there is no way to go from these infinite sums to a rational number.

$e = 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots$
$= \sum\limits_{n=0}^{inf} {\frac{1}{n!}}$

and

$\pi = 4 \sum\limits_{n=0}^{inf} {\frac{-1^n}{2n+1}}$

Again, you cannot put these into a nice rational form, but they are very good at approximating both. As a note to the OP, I think that you need to check common sense at the door. Common seems to tell you that no matter "how far you go" in the decimal expansion of .999..., you'll never get that extra "little bit" to make the expansion equal to 1. I think most people would agree that, at first glance, this makes sense. In fact, if you were to ask most people, they would probably say something similar.

However, in math, common sense has no place. Many, many, many times, common sense is dead wrong in math. This is especially true in real analysis (which is essentially what your question is about.) There are lots of "pathological" stuff in analysis. Things that don't seem to make any sense at all, but are, nonetheless, true.

Let me give you an example. There is something called the Cantor Set, C. This is a subset of the unit interval, I. Now, let I' denote the unit interval after we take the Cantor Set out of it. That is, in set notation, I' = I \ C. Additionally, $I' \cap C = \{\}$ and $I' \cup C = I$.

There are some interesting properties. For example, the length of $I'$ is 1. The number of elements in $I'$ is the same as the number of elements in $I$. So, you might think that C is just the empty set. Since I haven't defined the Cantor Set, then the empty set is certainly a possibility. However, the Cantor Set is not empty. In fact, it has the same number of elements as the unit interval and the same number of elements as $I'$. And, the length of the Cantor set is 0. So, as you can see, there is off stuff that just blows common sense out of the water. This .999...=1 thing is just the tip of the iceberg.

 

[Hurkyl]

I think you're taking the wrong lesson from counterexamples. A lot of time, a "pathological" example of X is not a demonstration that you have poor intuition about X, but that you were actually intuiting some other thing Y.

For example, when people are boggled about a facts like the Cantor set having the same number of elements as the entire interval, often it's because they are thinking about their intuitive notion of geometry, rather than a notion of cardinality.


I've read that on 0.999... specifically, people have problems because they flat out aren't thinking about a "number whose decimal expansion has infinitely many nonzero digits". They are thinking about things like a process of starting with "0." and adding "9"s one at a time, or they are thinking about a numeral with an unspecified large number of "9"s. Many of their misconceptions about 0.999... are, in fact, reasonable or even true statements about what they're really thinking about -- but they are firmly rooted in the land of confusion because they think they're thinking about 0.999...

 

[disregardthat]

@Robert: Elementary analysis and topology is often ridden with counter-examples, but they all share a common purpose: to show and build confidence in exactly why we need the at first glance seemingly unnecessary conditions for our theorems, and how horribly wrong it goes if we ignore them. Instead of boggling your mind with paradoxes (if you let them), you should rather let them teach you to keep your eye on the details, because they are always there for a reason. You can learn when to trust your intuition.

 

[Robert1986]

I think you're taking the wrong lesson from counterexamples. A lot of time, a "pathological" example of X is not a demonstration that you have poor intuition about X, but that you were actually intuiting some other thing Y.

For example, when people are boggled about a facts like the Cantor set having the same number of elements as the entire interval, often it's because they are thinking about their intuitive notion of geometry, rather than a notion of cardinality.


I've read that on 0.999... specifically, people have problems because they flat out aren't thinking about a "number whose decimal expansion has infinitely many nonzero digits". They are thinking about things like a process of starting with "0." and adding "9"s one at a time, or they are thinking about a numeral with an unspecified large number of "9"s. Many of their misconceptions about 0.999... are, in fact, reasonable or even true statements about what they're really thinking about -- but they are firmly rooted in the land of confusion because they think they're thinking about 0.999...

I don't think I mentioned "intuition", and if I did, it was a mistake. I used the term "common sense" which I consider to be very different from the notion of "intuition". To me, common sense is just a set of very shallow ideas that the "common man" has. For example, common sense would lead one to conclude that if you take some points away from a set X, then set X will have fewer points than it had to begin with. Of course, this is shallow and it doesn't take long to come up with an example to convince "the man on the street" of his errors. Intuition, on the other hand, is something that is developed by studying a certain topic. It is something that increases and changes as you learn more about whatever it is you are studying.

My point to the OP was that you have to accept the fact that some of the things you think might not be correct. There are some things that a person might think are true, but the math says that he is wrong. The Cantor Set is, IMO, a prime example of this.


And, as for the cause of people's misconceptions about .99..., I think what you wrote is pretty much what I described. People think that there is a finite number of 9's after the decimal place and thus the expansion "never makes it" to 1.

 

[Robert1986]

@Robert: Elementary analysis and topology is often ridden with counter-examples, but they all share a common purpose: to show and build confidence in exactly why we need the at first glance seemingly unnecessary conditions for our theorems, and how horribly wrong it goes if we ignore them. Instead of boggling your mind with paradoxes (if you let them), you should rather let them teach you to keep your eye on the details, because they are always there for a reason. You can learn when to trust your intuition.

I'm not sure what to make of this post. I don't let paradoxes "boggle" my mind. My point was to the OP and my point was that analysis has a lot of stuff that might seem incorrect at first glance, but are nonetheless true. Therefore, when confronted with a proof and several augments for why .99...=1, he should, as I said, check his common sense at the door.

 

[Char. Limit]

0.999999 is in the middle ground. So is 0.9999999 if the former didn't have enough 9's for you.

There is no middle ground between 0.999... and the set of all partial sums, though. This is a rather important geometric property of the number line.

Also, there is the sequence of numbers {1 - 10ⁿ} for those people who need to feel the need to consider a sequence of increasingly good approximations to 0.999...

And, of course, there is non-standard analysis for people who want a hyperfinite number of 9's, but still have a number that is only infinitessimally different from 0.999...
(hyperfinite for the non-standard measure of finiteness)


There is mathematical language for all sorts of ideas, and if there's not, it could be invented. The only true obstacle is when someone insists on grounding their reasoning firmly in the realm of vagueness and imprecision.

I don't quite understand the concept of "hyperfinite". Is it some sort of concept between the concepts of finite and infinite? What exactly do you mean when you say a "hyperfinite number of 9's"?

 

[Hurkyl]

(Did you see post #84?)

The non-standard model has all of the same objects, notions, constructions, and what not as the standard model does. In particular, it has its own notion of finiteness.

Since we often want to consider both the standard and non-standard notions of finiteness, it helps to use different words for them. So we continue with the tradition to prefix the non-standard version with "hyper".

For any positive hypernatural number H, the set of all integers between 0 and H is a hyperfinite set. However, this set is finite if and only if H is a standard natural number.

The non-standard model has its own version of cardinality. Any hyperfinite set has non-standard cardinality equal to a hypernatural number.

 

[Calrid]

I don't quite understand the concept of "hyperfinite". Is it some sort of concept between the concepts of finite and infinite? What exactly do you mean when you say a "hyperfinite number of 9's"?

It's a made up invented term for things that can supposedly exist beyond merely infinite number sets in sets, rather a semantic issue of no real importance, a simple infinite limit will always do in maths. I wouldn't concentrate too much on the details, as this thread highlights mathematicians cannot even conceive of an infinity any more than they can of what would be beyond such a beast and what properties such an invisible unicorn might have, perhaps a shade of pinkness? They just like to play with ghosts of what might be if reality and physical existence was different. It's like fairy stories, imaginary stuff that has no real practical use outside of maths in and of itself.

Hyperfine numbers can be relatively useful (well outside of science or applied maths where infinities are somewhat problematic) however there are some pretty dubious cardinality issues with transfinite numbers. Just grasping what an infinity could be is enough for most people to barely comprehend, like the OP. Mathematicians would beg to differ they have visualised something that is by axiom more than just infinite, but meh it's what they do, whether it is philosophically apt to make up ideas of beyond something that can never be reached is a matter of debate only outside of fortress maths. You'll never convince a mathematician that anything he says is epistemologically unjustifiable when all he needs is an axiom. It is true because I say it is, constructive logic and proof is irrelevant as is utility.

As far as transfinite systems go though, even though this has been a firm contention in maths since its inception, amongst the great minds of both maths and philosophy, whom it appears stand in corners according to whether they study philosophy or maths. It is apparently kinda illegal to discuss this subject as a system of "numeration" that will never have any use to anything outside of maths; being called into question is apparently so unsettling that it causes threads to be locked despite the great derth of material on this subject from all sorts of great minds from ancient times to today. It is not appropriate to question what is beyond "God". Perhaps if students of maths were to question their own axioms the whole number system would fall into chaos.

Perhaps definitions that make any sense are not important. Who knows..?

I of course disagree and find such silly blanket bans and contentions, with what are clearly circular and espistemologically dubious axioms, that are completely non constructive, and cannot define there own terms without resort to allusion, to be rather a nuisance to those who really want to understand this concept for what it actually is rather than what someone who didn't really understand what Cantor said thinks it is. It is not really an infinity it is trying to make the concept have utility beyond its definition which is rather what pure mathematicians do, when they run out of numbers that can exist, they make up ones that can't and then claim there are numbers beyond even that so on forever to infinity and beyond. Which is fine as long as like Cantor you make the proviso that these are not actual infinities, these are not really infinity, these are just conceptual things we use the title of infinity for, for want of a better word.

An infinity is as many of the people in this thread have said is not a number and applying mathematical arguments (whether you want to call it sets of infinite sets or infinity x infinity or even infinity^infinity to what is an undefined value is worthless beyond semantic arm waving. Outside of infinity has no logical, metaphysical or intrinsic worth to anything, it is and always has been all there is unbound. I'm not afraid to say that because it happens to be true and axiom is no substitute for logic.

If you divide the universe into infinite pieces, if you take infinite universes and divide them into infinite pieces, all you really have done is quantified the infinite in both cases as the same thing, there is nothing more than everything unbound, at least that has any utility or ever will.

 

[micromass]

As a mathematician, I don't think I quite agree with your analysis Calrid.

It's a made up invented term for things that can supposedly exist beyond merely infinite number sets in sets, rather a semantic issue of no real importance, a simple infinite limit will always do in maths.

Hyperfinite numbers were invented by Newton and Leibniz (and before) to give sense to their integral calculus. Sadly enough, the foundations of hyperreal where screwed up, and to fix it, they started to work with limits and epsilon-delta definitions. It is only in the later years, that infinitesimal quantities have found a real foundation. It's not because pure mathematicians wanted to invent something new, it's because they wanted to give a foundation to already existing stuff.

I wouldn't concentrate too much on the details, as this thread highlights mathematicians cannot even conceive of an infinity any more than they can of what would be beyond such a beast and what properties such an invisible unicorn might have, perhaps a shade of pinkness? They just like to play with ghosts of what might be if reality and physical existence was different. It's like fairy stories, imaginary stuff that has no real practical use outside of maths in and of itself.

So infinitesimals have no real practical use? Tell that to the physicists who work with infinitesimals every day. Entire physical theories are built up on the concept of infinitesimals. All mathematics wants to do is to give a foundation to them. This can be done in terms of hyperreals or differential geometry. But I don't think it's fair to call them fairy stories, imaginary stuff and useless.

Hyperfine numbers can be relatively useful (well outside of science or applied maths where infinities are somewhat problematic) however there are some pretty dubious cardinality issues with transfinite numbers.

I'd like to know what you mean with this. You mean the continuum hypothesis? That's not a problem of the transfinite numbers, but of the axioms of mathematics itself. Better axioms could resolve a lot of issues. (Although Godel proved that you cannot choose axioms that resolve all).

Just grasping what an infinity could be is enough for most people to barely comprehend, like the OP. Mathematicians would beg to differ they have visualised something that is by axiom more than just infinite, but meh it's what they do, whether it is philosophically apt to make up ideas of beyond something that can never be reached is a matter of debate only outside of fortress maths. You'll never convince a mathematician that anything he says is epistemologically unjustifiable when all he needs is an axiom. It is true because I say it is, constructive logic and proof is irrelevant as is utility.

Hmmm, you're the first to say that proof is irrelevant to mathematicians...
And transfinite numbers have not been invented because mathematicians thought they were fun. They were invented for a reason. Indeed, Cantor invented transfinite numbers to give sense to Fourier series. And you wouldn't call Fourier series useless do you?
Another big application of transfinite numbers is in probability theory, where the concept of sigma-algebra is fundamental.

As far as transfinite systems go though, even though this has been a firm contention in maths since its inception, amongst the great minds of both maths and philosophy, whom it appears stand in corners according to whether they study philosophy or maths. It is apparently kinda illegal to discuss this subject as a system of "numeration" that will never have any use to anything outside of maths; being called into question is apparently so unsettling that it causes threads to be locked despite the great derth of material on this subject from all sorts of great minds from ancient times to today. It is not appropriate to question what is beyond "God". Perhaps if students of maths were to question their own axioms the whole number system would fall into chaos.

Now you're just making things up. If you would know how real math works, then you would know that the axioms are being questioned every single day. And a student who does not question the axioms of mathematics, is not a good student in my opinion. Calling into question the axioms leads to very fruitful theories, like non-Euclidean geometry and the New Foundations theory. If people propose a new axiomatic system for a mathematical object, then I don't think any mathematician would hesitate to accept it if it were useful.

And as for the threads being locked. I have no qualms in discussing 0.999... and division by zero, if the OP was willing to learn. If somebody with a lot of knowledge about mathematics were to discuss these issues, I would listen and discuss with him/her. But you can't expect us to discuss something like this with somebody who hasn't seen limits and who still thinks that all mathematicians are wrong. If you do not grasp limits, then you have no idea what this question is even about.

In fact, I myself, have once constructed a new system where 1 does not equal 0.999... But the problem was that this system was ugly and not very useful. But don't tell us that we are not willing to change the axioms, because we are. The problem is often that the proposed new axioms do not deliver a nicer theory, on the contrary,...

Perhaps definitions that make any sense are not important. Who knows..?

So, which definitions do you think make no sense?

I think I've said everything I wanted, so I'll stop here. The only things that I want to make clear that mathematicians do not make things up for their amusement. There is often a need to understand something physical/mathematical/philosophical, and this is where the mathematical theories come from.

 

[Calrid]

As a mathematician, I don't think I quite agree with your analysis Calrid.
Hyperfinite numbers were invented by Newton and Leibniz (and before) to give sense to their integral calculus. Sadly enough, the foundations of hyperreal where screwed up, and to fix it, they started to work with limits and epsilon-delta definitions. It is only in the later years, that infinitesimal quantities have found a real foundation. It's not because pure mathematicians wanted to invent something new, it's because they wanted to give a foundation to already existing stuff.
So infinitesimals have no real practical use? Tell that to the physicists who work with infinitesimals every day. Entire physical theories are built up on the concept of infinitesimals. All mathematics wants to do is to give a foundation to them. This can be done in terms of hyperreals or differential geometry. But I don't think it's fair to call them fairy stories, imaginary stuff and useless.
I'd like to know what you mean with this. You mean the continuum hypothesis? That's not a problem of the transfinite numbers, but of the axioms of mathematics itself. Better axioms could resolve a lot of issues. (Although Godel proved that you cannot choose axioms that resolve all).
Hmmm, you're the first to say that proof is irrelevant to mathematicians...
And transfinite numbers have not been invented because mathematicians thought they were fun. They were invented for a reason. Indeed, Cantor invented transfinite numbers to give sense to Fourier series. And you wouldn't call Fourier series useless do you?
Another big application of transfinite numbers is in probability theory, where the concept of sigma-algebra is fundamental.
Now you're just making things up. If you would know how real math works, then you would know that the axioms are being questioned every single day. And a student who does not question the axioms of mathematics, is not a good student in my opinion. Calling into question the axioms leads to very fruitful theories, like non-Euclidean geometry and the New Foundations theory. If people propose a new axiomatic system for a mathematical object, then I don't think any mathematician would hesitate to accept it if it were useful.

And as for the threads being locked. I have no qualms in discussing 0.999... and division by zero, if the OP was willing to learn. If somebody with a lot of knowledge about mathematics were to discuss these issues, I would listen and discuss with him/her. But you can't expect us to discuss something like this with somebody who hasn't seen limits and who still thinks that all mathematicians are wrong. If you do not grasp limits, then you have no idea what this question is even about.

In fact, I myself, have once constructed a new system where 1 does not equal 0.999... But the problem was that this system was ugly and not very useful. But don't tell us that we are not willing to change the axioms, because we are. The problem is often that the proposed new axioms do not deliver a nicer theory, on the contrary,...
So, which definitions do you think make no sense?

I think I've said everything I wanted, so I'll stop here. The only things that I want to make clear that mathematicians do not make things up for their amusement. There is often a need to understand something physical/mathematical/philosophical, and this is where the mathematical theories come from.

I never said limits weren't useful an hence infinitesimals are useful if we accept infinity cannot ever be equalled only approached we must also accept that nothing cannot be represented physically but can only be approached. It isn't limits that are the problem or even hyper reals, it's transfinities, what it means to have infinite infinite sets where the problem becomes epistemologically inexplicable. Which rather makes the rest of your arguments redundant at least if you mean anything that is bound to a limit like calculus etc. For example is pi closer to pi at aleph 0, infinity in natural numbers, or is it closer to infinity at aleph 1 or aleph 2, or aleph omega? What does it mean to set up limits that are more than infinite or less? is it conceptually viable, will what is beyond reality ever have utility unless imagination is of course just a part of the set that exists.

Does the photon have 0 mass, or have we only measured it to a lower bound to which the difference is practically inconsequential?

I acknowledged hyperreals have utility in pure calculus issues. What I don't acknowledge is that beyond infinity ever could make any sense to anyone. What is beyond that which we cannot even imagine except sophistry and religious fervour or fairy tales?

You have no idea what value infinity has, and like wise you have no idea how to cardinalise an actualy infinite value, because you could never reach its limit. This may allow us to say that infinity ^ infinity is akin to aleph omega, but this actually means nothing, nor ever could. It is eternally philosophical arm waving. It does not actually mean infinity, unless we make the destinction between something like the size of the universe, a countable infinity and infinity a number in which no matter how long one spent trying to approach it, one would never reach it. It is beyond definition. To define it is as many philosophers have said is to define God: that which cannot be comprehended or defined. So what is beyond that which is beyond all that exists exactly, and why should we care?

I agree that .999... = 1 at infinity but that is only the case if we do not use transfinities, otherwise it is more or less equal depending on what set you are using. Can you see why such mental masturbation is useless? We only need one limit for any proof in any field of maths you care to name, we can derive all the rules of maths from simply having infinity as 1 asymptotic non defined value. To be honest we can probably get away without limits in most of maths with the exception of course of calculus and set/ number theory which itself underpins mathematical axioms. Science it doesn't even get a mention as its physically impossible. It is a very useful and purposeful limit when it is undefined.

The threads that were locked were in philosophy and general. I don't have an issue with that although an explanation would of been nice, particularly when I requested one. You know like thread locked pending moderation is not really an explanation..? But meh whatever.

I was being sarcastic about mathematicians questioning axioms you'll note also hence the smilie.

The only real axiom that makes sense I think is how can I apply this to reality, how might I use this: beyond all that exists? I guess that is where applied mathematicians and mathematicians differ. Cantors continuum is not even a non constructive proof, it cannot even define its terms as they are indefinite by every axiom outside of that one. Not that I argue with: if that given axiom is accepted without question then it must be true, but axioms don't need to be deductive or require proofs they just need to be accepted. I don't think we should accept that axiom because it has no utility or function and it cannot be iteratively proven only alluded to. As Kant said existence is not a predicate, by which he meant nothing exists just because it has a property we can imagine. It certainly doesn't exist logically, except as a limit to reality, if we cannot even comprehend it, much less what lies beyond it has or could have any utility to anything except circular self referential a priori assumptions.

I don't think mathematicians actually gained any real further understanding from imagining what infinite infinities might be simply because they cannot even comprehend an infinity in the first place without making it something it is not. The universal set on its own would be enough to define all that is to which all sets are part of, it and all mathematical branches from topology to algebra likewise can be contained in a set of definitive values, not illusory ones or not much better allusory (is that a word) ones. Sure it's a semantic issue, but aren't semantic issues sometimes very important?

 

[micromass]

I never said limits weren't useful an hence infinitiessimals are useful if we accept infinity cannot ever be equalled only approached we must also accept that nothing cannot exist but can only be approached. It isn't limits that are the problem or even hyper reals, it's transifnities, what it means to have infinite infinite sets where the problem becomes epistemologically inexplicable. Which rather makes the rest of your arguments redundant at least if you mean anything that is bound to a limit. For example is pi closer to pi at aleph 0, infinity in natural numbers, or is it closer to infinity at aleph 1 or aleph 2, or aleph omega? What does it mean to set up limits that are more than infinite or less? is it conceptually viable, will what is beyond reality ever have utility unless imagination is of course just a part of the set that exists.

I can't really make any sense of this. What have limits to do with transfinite sets? A transfinite set is just a process that continuous to infinity. It's a very useful concept in mathematics and physics.

Who cares whether infinite sets exist in real life? That's not the problem here. We didn't invent transfinite numbers to represent anything existing. We invented transfinite numbers to give a certain foundation to something.

I acknowledged hypereals have utility in pure calculus issues. What I don't acknowledge is that beyond infinity ever could make any sense to anyone. What is beyond that which we cannot even imagine except sophistry and religious fervour or fairy tales?

This is where you're wrong. We can comprehend infinity. It's one of the major feats of the last century: that infinity finally makes sense to us! We can calculate with infinite sets, we can present a foundation to many argument, etc.

You have no idea what value infinity has, and like wise you have no idea how to cardinalise an actualy infinite value, because you could never reach its limit.

Sure, we can. $$\mathbb{N}$$ is an actual infinite value, and we can easily cardinalise it as $$\aleph_0$$. And again, I fail to see what transfinite numbers have to do with "limits".

This may alow us to say that infinity ^ infinity is aleph omega, but this actually means nothing, nor ever could. It is eternally philosophical arm waving. It does not actually mean infinity, unless we make the destinction between something like the size of the universe, a countable infinity and infinity a number in which no matter how long one spent trying to approach it, one would never reach it. It is beyond definition. To define it is as many philsophers have said is to define God. So what is beyond that which is beyond all that exists exactly, and why should we care?

Oh please, just because many philosophers say it is impossible, doesn't mean that it is impossible. 1000 years ago they said we could never step on the moon, and behold: we did it. Likewise, they said we could never comprehend infinity: but then Cantor invented his transfinite numbers to give a representation to infinite values.

Infinity is well understood by mathematicians nowadays. It's one of the most beautiful things about mathematics: that abstract notions can serve as an aid to understand something as abstract as infinity!

I agree that .999... = 1 at infinity but that is only the case if we do not use transfinities, otherwise it is more or less equal depending on what set you are using.

I seriously did not understand this statement... 0.999...=1 at infinity? What does that even mean? What does this have to do with transfinities?

Can you see why such mental masturbation is useless? We only need one limit for any proof in any field of maths you care to name, we can derive all the rules of maths from simply having infinity as 1 asymptotic non defined value.

OK, just because you're using fancy terms like "asymptotic non defined value" or "mental masturbation", doesn't mean that you're right. Can you please explain to a simple mathematician such as me, what you mean exactly?

You seem to have a problem with transfinite numbers, that's clear. But I don't see which one. We never said that transfinite numbers occur in nature, did we? In fact, I'm a strong believer that the universe is finite. However, mathematical infinites just makes our life easier and it offers an accurate approximation to a lot of mathematical things.

The real numbers don't exist in real life, in fact, when doing physics, we could be ok with just rational numbers: indeed, every measurement we can possibly do is rational. However, we work with real numbers because it simplifies a lot and because it's a reasonable approximation to our measurement. Who cares whether they exist in real life, that's not what this thing is about!

 

[micromass]

As you yourself say the only real axiom that makes sense I think is how can I apply this to reality, how might I use this beyond all that exists? I guess that is where applied mathematicians and mathematicians differ. Cantors continuum is not even a non constructive proof, it cannot even define its terms as they are indefinite by every axiom outside of that one.

Transfinite numbers are well-defined. So I don't see your point. Transfinite numbers can even occur in constructive mathematics: I can give a good definition for $$\aleph_0$$ if I want to. If things weren't well defined, then mathematicians would be the last to use them.

Not that I argue with: if that given axiom is accepted without question then it must be true, but axioms don't need to be deductive or require proofs they just need to be accepted.

Axioms are always true in the sense that: if a system satisfies the axioms, then it satisfies all theorems coming from the axioms. For example, if a set satisfies the group axioms, then it satisfies all the theorem that follow from the group axioms.

Mathematics is an "if-then" science. We must always check IF the axioms are satisfied, and THEN we can apply the consequences. Mathematicians never state that their axioms relate to real life. That's the physicist's job. And fortunately, most axioms DO relate to real life!

I don't think we should accept that axiom because it has no utility or function and it cannot be iteratively proven only alluded to. As Kant said existence is not a predicate, by which he meant nothing exists just because it has a property we can imagine. It certainly doesn't exist logically, except as a limit to reality, if we cannot even comprehend it, much less what lies beyond it has or could have any utility to anything except circular self referential a priori assumptions.

I don't understand this. Can you please use some easier terms. You're talking with a simple math-guy here. Not with a fancy philosopher...

 

[Calrid]

Have to go I, will answer later, just hope this thread isn't locked because I take issue with the axioms as they are stated. Any more than someone would of locked Hilbert's thread on his hotel. :tongue:

Transfinite numbers are well-defined. So I don't see your point. Transfinite numbers can even occur in constructive mathematics: I can give a good definition for LaTeX Code: \\aleph_0 if I want to. If things weren't well defined, then mathematicians would be the last to use them.

I will say this though can you show me an example of an infinite number, let alone a transfinite one without resorting to axioms about it having some property you couldn't really imagine given it is just a predicate? God exists because he is the greatest thing that I can imagine is the same argument aka the ontological argument, it is just repackaged in philosophical terms. It is equally as weak and depends on an a priori assumption, ie an unprovable axiom based on an indefinite quality.

So then what is beyond all that exists is it God or is it actually something we can know or even imagine? What utility then do such infinite "Gods" have to anything beyond limits to the conceivable or reality?

 

[micromass]

I will say this though can you show me an example of an infinite number, let alone a transfinite one without resorting to axioms about it having some property you couldn't really imagine given it is just a predicate?

I don't understand the issue you have with axioms. Certainly you need axioms to do mathematics. If you have no axioms or definitions, then there is nothing you can do. You can't even show that there exists anything then!
Mathematics is the "science" that works with axioms, and proves things from that axioms. It is always correct because it works with conditional (i.e. if-then) statements: IF the axioms are correct THEN this is true.
Showing that the axioms actually hold is something for physicists. And often, the axioms that are being considered in mathematics form a reasonable approximation with reality.
Every science works with axioms actually. In physics, these axioms are being given by experiments.

And yes, I can easily give an example of a transfinite number: $$\mathbb{N}$$. This is a transfinite number. Nobody cares whether the naturals exist in real life. We work with them because it is an approximation to reality.

God exists because he is the greatest thing that I can imagine is the same argument aka the ontological argument, it is just repackaged in philosophical terms. It is equally as weak and depends on an a priori assumption, ie an unprovable axiom based on an indefinite quality.

This is not a good analogy. The "God exist because there is nothing greater"-argument fails because there is no way to represent this in the mathematical language. Transfinite numbers CAN be represented in mathematical language. And therefore, it is correct!

So then what is beyond all that exists is it God or is it actually something we can know or even imagine? What utility then do such infinite "Gods" have to anything beyond limits to the conceivable or reality?

Why do you say that infinities are not conceivable. I can very easily imagine something infinite. And my mathematics allows me to work with infinite things.
That you say that they are not realistic is another thing. But again: no mathematician or physicists cares whether what they're doing is realistic. As long as the outcome conformes to the the reality: and it does!