I would like to argue about .999

[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.

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!

 

[Caramon]

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

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

.9 + .09 + .009 + .0009 + .00009 + ... = .9 \sum_{n=0}^\infty \left(\frac{1}{10}\right)^n

Now, evaluating the sum on the right, we use the fact (proven below) that...

\sum_{n=0}^\infty r^n = \frac{1}{1-r}

for all r with a magnitude less than 1. Using this fact, we find that...

.9 \sum_{n=0}^\infty \left(\frac{1}{10}\right)^n = .9 \frac{1}{1-.1} = \frac{.9}{.9}=1

Now, to prove that fact that we used, note the proof below:

S = \sum_{k=0}^{n-1} a r^k = a + a r + a r^2 + a r^3 + a r^4 + ... + a r^{n-1}

rS = a r + a r^2 + a r^3 + a r^4 + a r^5 + ... + a r^n

S - rS = a - a r^n = a (1 - r^n)

S(1-r) = a (1 - r^n)

S = \frac{a (1-r^n)}{1-r}

Now let n go to infinity. For r with a magnitude less than 1, r^n tends to 0 as n tends to infinity. Thus...

\lim_{n \rightarrow \infty} S = \lim_{n \rightarrow \infty} \frac{a (1-r^n)}{1-r} = \frac{a}{1-r}

Q E D

/thread

This thread is INFINITELY hilarious because people don't need to say anything other than, "oh wow this proof is great."

 

[Hurkyl]

Since the OP is no longer posting in the thread and these last few posts don't really look like mathematics, I think it's time to close this thread.

Calrid: if you want to post your ideas in the math subfora here, you're going to have to be clear and precise. For example, no making up an idea like one number equaling another number "at infinity" unless you first define what you mean (or at least make a reasonable attempt).

While I normally like philosophical discussions about math, they aren't very useful when they are overly vague/imprecise or one side appears to assert factually incorrect statements, even going so far as to explicitly refuses to acknowledge reality. (e.g. the reality that "beyond infinity" can and does make sense to some people. I assume from the context that you are referring to one infinite thing being larger than another -- but there are other cases where beyond infinity makes sense as written, such as the ideal points^* of hyperbolic geometry)

*: This might be the wrong name for them -- I'm having trouble finding a reference. For those who know hyperbolic geometry, I'm referring to the extension where any pair of distinct non-parallel lines meet in two points.

 

[HallsofIvy]

No, "ideal points" is correct. Although I would call them "points at infinity" rather than "beyond" infinity.

 

[Hurkyl]

No, "ideal points" is correct. Although I would call them "points at infinity" rather than "beyond" infinity.

Hrm. I would have expected the phrase "at infinity" to refer only to the boundary of the set of ordinary points, rather than including the points on the other side of the boundary. Then again, I honestly don't recall the last time I've seen "at infinity" used in a situation where the space of ordinary points has a nonempty exterior.

 

[HallsofIvy]

Okay, I mis-interpreted what you said. The "points at infinity" are the "ideal points" and the points "beyond" infinity are "ultra- ideal" points.