Consecutive Reals: A Hypothetical Possibility?

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In summary, the conversation discusses the concept of consecutive real numbers and whether they can be defined or exist in the set of real numbers. One perspective argues that there is no intuitive way to refer to the "next" real number and that the existence of consecutive reals is independent of the ability to "move along" the real number line. Another perspective suggests that, assuming the Axiom of Choice, there is a successor-like function for every set. The conversation also touches on Zeno's Paradox and the issue of defining successive reals using time. Ultimately, the question remains whether consecutive reals are a justified, if not expressible, idea.
  • #1
hddd123456789
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I'm just thinking aloud on this one. So let's imagine a hypothetical function S_r that's the real analogue of S_n (the successor function over the naturals). Now let's say a, b in R are defined as follows: b=S_r(a); b is the real successor of a. Now of course if such a function existed then b-a must equal zero. If it didn't, then we can always find a c in R that's between a and b which would then make c a new candidate for a's successor. But if a-b is equal to zero, then a must necessarily equal b and so b can't be its successor. Such an argument seems to show that consecutive reals cannot exist, or at least can't be expressed using real numbers themselves.

On the other hand, if we are working with the reals under the assumption that consecutive reals do not exist, then there is no intuitive way to refer to the "next" real number. Certainly there's no way to define it using reals themselves as above. But if one says that there actually and certainly is no next real number then how are we moving along the field at all? How do you move between one and two if there's no conceptual way of referring to the real number that comes immediately after the number 1 in the set of reals. If b-a must equal zero then you can add all the zeroes you want in the world to a but it won't change a thing unless you give infinity some mystical power to turn those zeroes into real values.

I'm somewhat familiar with Cantor's diagonal argument now and I understand that that is precisely the whole point: that there exists no bijection between the reals and the naturals, and so no consecutive real by definition. But while I submit that such a thing can't be formalized in anyway using current language, doesn't the above reasoning at least justify the existence of consecutive real numbers in some as of yet to-be-defined form?
 
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  • #2
hddd123456789 said:
But if one says that there actually and certainly is no next real number then how are we moving along the field at all? How do you move between one and two if there's no conceptual way of referring to the real number that comes immediately after the number 1 in the set of reals. If b-a must equal zero then you can add all the zeroes you want in the world to a but it won't change a thing unless you give infinity some mystical power to turn those zeroes into real values.

Is this not a restatement of Zeno's Paradox?
 
  • #3
A couple of points:

You can choose an ordering on the real numbers which is different from the one we normally give it such that you can in fact have a well defined successor function for each real number. This ordering is probably terrible and doesn't correspond to what we think of when we think of real numbers being larger or smaller than each other.

That said, the problem of defining successive reals is independent of the ability to "move along" the real number line. What does that mean mathematically? Most people think of it as "I give you a position for each point in time". Time typically takes any real number as a value, so to say "where am I at the next point in time" is a meaningless question to begin with. More generally you will be hard pressed to give a good definition of moving along the real number line such that an actual contradiction can exist.
 
  • #4
1MileCrash said:
Is this not a restatement of Zeno's Paradox?

Yeah that occurred to me actually, just wasn't completely sure as I have just a touch-and-go familiarity with it. I know that this really sinks back some ways to that basic question which has been solved, for all practical purposes, by calculus. So this is really more of a, I suppose, metaphysical question: are consecutive reals a justified, if not expressible, idea?
 
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  • #5
The phenomenon that you are investigating is more related to the fact that the normal order on the reals is what is called a dense linear order. It's the dense part that makes it seems that there is no "successor-like function". The rationals are also densely ordered, and so there is no function on the rationals which is successor-like and respects the normal order. So cardinality (and thus Cantor's diagonal argument) is really not the issue.

It turns out that, assuming the Axiom of Choice/Well-Ordering Principle, there is a successor-like function for every set. This isn't so hard believe to if you believe that every set has a cardinality in the sense that it can be put into 1-1 correspondence with a cardinal number.
 
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  • #6
Office_Shredder said:
That said, the problem of defining successive reals is independent of the ability to "move along" the real number line. What does that mean mathematically? Most people think of it as "I give you a position for each point in time". Time typically takes any real number as a value, so to say "where am I at the next point in time" is a meaningless question to begin with. More generally you will be hard pressed to give a good definition of moving along the real number line such that an actual contradiction can exist.

That depends of course on what you mean by "moving along" the real line. I'm referring to the intuitive idea encompassed in the "intermediate value theorem" that if you are moving from one to two, you must "move along" all the reals between 1 and 2 to get to 2. You can't be jumping around seemingly random reals using some arbitrary mapping with the naturals. And the problem I have with using time is that it is a circular argument, much like defining finite vs infinite (i.e. we might define something as having a finite length if it takes a finite amount of time to measure it). So I could just as well ask how does one move through the field of time between the 1st and 2nd second if there is no conceptual framework validating the move to the next real after the 1st second.
 
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  • #7
gopher_p said:
It turns out that, assuming the Axiom of Choice/Well-Ordering Principle, there is a successor-like function for every set. This isn't so hard believe to if you believe that every set has a cardinality in the sense that it can be put into 1-1 correspondence with a cardinal number.

I think I've seen it for the rationals, but I actually wasn't aware that there was such a discreet mapping for the reals. Even so, I guess I'm just defining S_r differently to mean what is encompassed in the intermediate value theorem; namely that to move from 1 to 2, you must move through all the reals between them starting at 1. Or if I have the correct terminology, there ought to be a real successor function (S_r) that is totally ordered to allow one to "move along" the field of reals.
 
  • #8
hddd123456789 said:
I think I've seen it for the rationals, but I actually wasn't aware that there was such a discreet mapping for the reals.

There is no mapping from the naturals that covers all of the reals. Cantor's proof still applies.

What is provable (given the Axiom of choice) is that there exists a "well ordering" of the reals. However, the proof of this is not constructive. That is to say that it is not possible to explicitly exhibit such an ordering.

An "order" (or sometimes "partial order") is a binary relation, "<" between pairs of elements that is transitive, irreflexive and anti-symmetric. Transitive means that if a < b and b < c it follows that a < c. Irreflexive means that is it never the case that a < a. Anti-symmetric means that if a < b it follows that it is not the case that b < a.

A "total order" is an order that also satisfies trichotomy. That is, given any pair of elements, either a < b, b < a or a = b.

A "well ordering" is a total order that also has the property that any non-empty set has a smallest member.

The natural numbers are well-ordered by their usual ordering.

The rational numbers are not well-ordered by their usual ordering. For instance, the set of all rationals has no smallest member. But if you map each rational to a natural number then the rational numbers are well-ordered by the order that is induced by that mapping.

The real numbers cannot be mapped (one to one) to the integers. So that trick does not work for them.

Even so, I guess I'm just defining S_r differently to mean what is encompassed in the intermediate value theorem; namely that to move from 1 to 2, you must move through all the reals between them starting at 1. Or if I have the correct terminology, there ought to be a real successor function (S_r) that is totally ordered to allow one to "move along" the field of reals.

That terminology is not apt. The difference between what "ought to be" and what "is" is often large. Note that if you are going to invoke some hypothetical well-ordering of the reals then the intermediate value theorem and the prerequisite notion of continuity become largely irrelevant -- those are tied to the standard ordering.

Given a well ordering...

For every point x "between" [in the sense of the well ordering] 1 and 2 there would be a "next point" [the smallest real that is greater than x and less than or equal to 2]. That's guaranteed by what it means to have a well order. But that doesn't mean that you could get from 1 to 2 a step at a time, even with infinitely many steps. It could be that some numbers "between" 1 and 2 are not successors of any other number.
 
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  • #9
jbriggs444 said:
Note that if you are going to invoke some hypothetical well-ordering of the reals then the intermediate value theorem and the prerequisite notion of continuity become largely irrelevant -- those are tied to the standard ordering.

Could you please expand on this? Particularly, how is the notion of continuity tied to standard ordering?

jbriggs444 said:
It could be that some numbers "between" 1 and 2 are not successors of any other number.

And is this a hypothetical or has this actually been shown to be the case?
 
  • #10
hddd123456789 said:
Could you please expand on this? Particularly, how is the notion of continuity tied to standard ordering?

How do you define continuity?

The definitions I am familiar with involve epsilons and deltas and "less than". How do you do "less than" without an ordering? If you change out the ordering, you've changed out the definition of continuity.
 
  • #11
Oh nvm, I should've looked up what standard ordering means first. But I still don't understand why invoking a well-ordering on the reals removes standard ordering. This is probably over my head, but I'll do my best to follow.
 
  • #12
jbriggs444 said:
Note that if you are going to invoke some hypothetical well-ordering of the reals then the intermediate value theorem and the prerequisite notion of continuity become largely irrelevant -- those are tied to the standard ordering.

Sadly, I'm not able to make heads or tails of this just yet (at least not using what definitions I could find for the terminology used), though it has given me the impetus to finally open this book on set theory I got some time back.

Having said that, are you essentially saying (in laymen's terms) that since I'm basically quantizing the field, I can't then base that quantized field on the concept of continuity, the definition of which requires that the field not be quantized?
 
  • #13
hddd123456789 said:
Having said that, are you essentially saying (in laymen's terms) that since I'm basically quantizing the field, I can't then base that quantized field on the concept of continuity, the definition of which requires that the field not be quantized?

Yes, this is correct. Continuity in particular requires the notion of an interval: take two numbers, and every number in between. This doesn't make sense once you consider a well-ordering of the reals (which says for every real there is a next real). The reason why is that the "next real number" is going to look really weird from our usual perspective, the first real number might be 0, then the "next one" 3, then the "next one" -2.4, then the "next one" 2e. And you lose the idea of real numbers being "really close" to each other - in the ordering I described above, the closest number to 0 is 3; every number is farther away in the ordering so talking about things like limits stops making sense.
 
  • #14
To find a formal disproof of the existence of a successor in the standard ordering, see, e.g., the (constructive) argument that between two Rationals there is a Rational; basically between any two Reals there is a Rational, and there is an Irrational number. Or look at the decimal representation of any two Reals, that differ in the n-th decimal place, you can also go farther back than the n-th place to find an in-between Real.
 
  • #15
Ok, please stop me if I'm just putting words in your mouths, but when I take this:

Office_Shredder said:
Yes, this is correct. Continuity in particular requires the notion of an interval: take two numbers, and every number in between. This doesn't make sense once you consider a well-ordering of the reals (which says for every real there is a next real).

together with this:

jbriggs444 said:
For every point x "between" [in the sense of the well ordering] 1 and 2 there would be a "next point" [the smallest real that is greater than x and less than or equal to 2]. That's guaranteed by what it means to have a well order. But that doesn't mean that you could get from 1 to 2 a step at a time, even with infinitely many steps. It could be that some numbers "between" 1 and 2 are not successors of any other number.

can't I interpret this as saying that a well-ordering implies the existence of consecutive reals (in the sense of there being a "next point" after 1), but there is no mapping to the naturals that can take you across a particular interval, or subset of the reals, in a step-by-step manner.
 
  • #16
WWGD said:
To find a formal disproof of the existence of a successor in the standard ordering, see, e.g., the (constructive) argument that between two Rationals there is a Rational; basically between any two Reals there is a Rational, and there is an Irrational number. Or look at the decimal representation of any two Reals, that differ in the n-th decimal place, you can also go farther back than the n-th place to find an in-between Real.

Yes if such a successor function were to exist for the reals, it couldn't be defined in terms of reals as that very quickly leads to the problems you mentioned and in the OP as well.
 
  • #17
hddd123456789 said:
can't I interpret this as saying that a well-ordering implies the existence of consecutive reals (in the sense of there being a "next point" after 1)

A next point that need not be anywhere near 1 in our normal sense of "near", yes.

but there is no mapping to the naturals that can take you across a particular interval, or subset of the reals, in a step-by-step manner.

A subset of the reals you might map one to one to the naturals. For instance, {1.0, 2.0, 3.0, etc} would be easy to map. A non-trivial interval you can't.
 
  • #18
jbriggs444 said:
A subset of the reals you might map one to one to the naturals. For instance, {1.0, 2.0, 3.0, etc} would be easy to map. A non-trivial interval you can't.

I appreciate you clarifying the terminology for me.

jbriggs444 said:
A next point that need not be anywhere near 1 in our normal sense of "near", yes.

Ok, sorry if this is getting repetitive, but as it's been proven that the reals are well-ordered, is it this proof in particular that rules out the possibility of constructing a standard order of the reals? Or is it ruled out by more intuitive notions that a standard ordering of the reals can't be described in any sensible way (i.e. standard-ordered reals in the interval (0,1) have no least element)?

I'm asking because I've been playing around with this idea of consecutive reals on-and-off (just realized I posted about them here 2 years ago) and I figured if there was anything to this, then it should be possible to infer their existence with available terminology. Though at this point I get the feeling that nothing short of directly challenging Cantor's arguments on infinity would cut it if one's trying to will such a concept into existence.
 
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  • #19
hddd123456789 said:
Ok, sorry if this is getting repetitive, but as it's been proven that the reals are well-ordered

A set is not "well-ordered". A set is well-ordered by an order. Or an order is a well-ordering of a set. The reals are not well-ordered. However, it is provable that there exists a well-ordering of the reals.

is it this proof in particular that rules out the possibility of constructing a standard order of the reals?

Given a model of the reals (e.g. as equivalence classes of Cauchy sequences, as Dedekind cuts or as canonical infinite decimals), the standard order is perfectly constructible. No well-ordering of the reals is constructible.

Or is it ruled out by more intuitive notions that a standard ordering of the reals can't be described in any sensible way (i.e. standard-ordered reals in the interval (0,1) have no least element)?

That just demonstrates that the standard ordering is not a well-ordering.
 
  • #20
jbriggs444 said:
A set is not "well-ordered". A set is well-ordered by an order. Or an order is a well-ordering of a set. The reals are not well-ordered. However, it is provable that there exists a well-ordering of the reals.



Given a model of the reals (e.g. as equivalence classes of Cauchy sequences, as Dedekind cuts or as canonical infinite decimals), the standard order is perfectly constructible. No well-ordering of the reals is constructible.



That just demonstrates that the standard ordering is not a well-ordering.

I'm going to stop embarrassing myself now in trying to use these terminologies -_- Just one last question, though I'm not sure if it needs to be in a separate thread/forum. Isn't the basis of Cantor's ideas on infinities, namely the existence of completed infinities, essentially a philosophical position rather than a provable theorem? The whole one-to-one correspondence idea makes sense and all, but it seems to me that such a one-to-one mapping only gives information on cardinality if one assumes the existence of these completed infinities.
 
  • #21
hddd123456789 said:
I'm going to stop embarrassing myself now in trying to use these terminologies -_- Just one last question, though I'm not sure if it needs to be in a separate thread/forum. Isn't the basis of Cantor's ideas on infinities, namely the existence of completed infinities, essentially a philosophical position rather than a provable theorem?

You are correct that the existence of infinite sets is usually taken as an axiom. It is not deducible from the other axioms of set theory.

A very great part of mathematics can be seen as a game of "what if". It is not so much that we claim that the axioms are "true" (whatever that might mean). It's that we explore their consequences anyway. Let the philosophers worry about whether there really is an infinite set. Cantor's theorems still follow from the axioms either way.

The whole one-to-one correspondence idea makes sense and all, but it seems to me that such a one-to-one mapping only gives information on cardinality if one assumes the existence of these completed infinities.

The definition of cardinality in terms of one to one correspondence works perfectly well for finite sets. It's the notion of counting sheep by putting a pebble in your bag for every sheep in the flock and then counting the pebbles. It's not very deep, but it does work.
 
  • #22
jbriggs444 said:
The definition of cardinality in terms of one to one correspondence works perfectly well for finite sets. It's the notion of counting sheep by putting a pebble in your bag for every sheep in the flock and then counting the pebbles. It's not very deep, but it does work.

Oh right, I meant in terms of infinite sets; that information about the cardinality of arbitrary infinite sets only seems derivable when one accepts the existence of the completed set of naturals to which one is attempting to map to.

And with regards to the following:

jbriggs444 said:
Let the philosophers worry about whether there really is an infinite set. Cantor's theorems still follow from the axioms either way.

I have been reading up on some background in Cantor's development of infinities and from what I've read so far, I found it interesting that rather than shy away from the philosophical questions of infinity, he seemed to have embraced them and made it an important part of justifying his work (esp. given the religious climate of the time). It seems to me that any serious critique of his work would have to confront his philosophy as much as the theorems derived from it.

He really thought of his numbers to be as "real" as the natural numbers and felt that his transfinites could be afforded, by their very nature, different properties than the finites. It's pertinent then to note that if his completed infinities exist, then surely by the very nature of something that completes, it must come to and "end". As he afforded these numbers novel properties, then I don't see why one can't imagine that such completed sets must "end", just not in the same way that finite sets end. I'm wondering about this as it pertains to a possible critique of Cantor's diagonal argument.
 
  • #23
hddd, I don't know what you mean by "completed" infinities. At any rate trying to apply a layman's definition of a word to a mathematical term is pointless, so thinking that if something is called completed it must have an "end" (whatever that is supposed to mean, since it hasn't been defined mathematically at all) is incorrect.
 
  • #24
Well that's just it, I'm not sure that "completed" really is a mathematical term but more a philosophical term assumed as an axiom of set theory. And by completed, I'm using it as I understood it from its use in the link provided in my above post. If the basis of completed infinities as employed by Cantor is philosophical in nature, then it subjects itself to philosophical arguments as above. Please correct me if I'm missing something here.
 
  • #25
I see now, they're just using that as another way of saying that the infinity is represented as an actual set. For example, is the set N of natural numbers an actual set (complete), or is it just something that you can imagine by taking larger and larger sets of natural numbers {1,2,...,n} and imagine what happens as n goes to infinity. Here the word complete is just used to meant "actually exists". It is philosophical in the sense that you can also assume the set of natural numbers is not a set that actually exists in mathematics and see what the consequences are of that. Saying that if it is a set it must have some "end" to it is where you stop making sense, because you pulled out the word "end" from the layman's definition of "complete" which is different from the meaning that it has when talking about sets.

Furthermore, this is a math forum, not a philosophy forum. If you want to talk about the philosophy of infinite sets it should be done in a different setting. We can talk about the consequences of different axiom choices but debating the philosophical value of them is beyond the scope of the forum.

As an aside the debate of "actual" (complete) versus "potential" infinities is something that is not really a debate, it would be good to keep in mind that there isn't a point in trying to argue for one versus the other. The mathematics of infinite sets is well understood and well-grounded at this point in axiomatic set theory.
 
  • #26
hddd123456789 said:
Isn't the basis of Cantor's ideas on infinities, namely the existence of completed infinities, essentially a philosophical position rather than a provable theorem? The whole one-to-one correspondence idea makes sense and all, but it seems to me that such a one-to-one mapping only gives information on cardinality if one assumes the existence of these completed infinities.

I have not read Cantor's papers but I think he uses a generalization of counting to define the ordinal numbers that are infinite. I believe that he then proves that one of these numbers is not of the same cardinality as the integers.

If one considers the well ordering of the ordinals that is generated in Cantor's "counting" then each ordinal is the end point of the process of counting that preceded it. So the number 5 would be the the end point of counting from one to 5, the first infinite ordinal would be the end of counting all finite ordinals, the first uncountable ordinal the end of counting all countable ordinals and so on.

I would be interested to know what exactly Cantor meant by "completed infinity." Would it be a Cauchy sequence together with its limit point?
 
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  • #27
Office_Shredder said:
Saying that if it is a set it must have some "end" to it is where you stop making sense, because you pulled out the word "end" from the layman's definition of "complete" which is different from the meaning that it has when talking about sets.

Furthermore, this is a math forum, not a philosophy forum. If you want to talk about the philosophy of infinite sets it should be done in a different setting. We can talk about the consequences of different axiom choices but debating the philosophical value of them is beyond the scope of the forum.

I do understand your point about forcing a layman's definition, though I can't say I meant it to be as non-mathematical as that. I guess what I meant is something along the lines of the image I attached (I don't have much experience writing Aleph_0). I'm sure there's some of what's not considered to be good math in there, and I'm not even really sure what point I can make using this. Though I was thinking something along the lines of: with respect to the set of reals with elements x_n, x_c is a "hyper-real" in the sense that it has sqrt(2) as many digits as any x_n. This is how I meant to interpret the notion of completed sets; that if they are completed, then shouldn't I be able to consider the geometrical figure that the diagonal argument produces?

But I guess for this to have any meaning, the existence of these "hyper-reals" would have to be shown along with somehow showing that general statements like "there are more reals between 0 and 1 than the natural numbers" can't be made without qualifying which "degree" of reals [Edit: and naturals] we are referring to? I'm just feeling my way through the range of possibilities, pardon if the thread is veering off course.
 

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  • #28
As I read that drawing, you are considering that the "length" of the Cantor's diagonal is sqrt(2) times the number of rows and also sqrt(2) times the number of columns.

But that's nonsense. Clearly there exactly as many digits on the diagonal as there are rows. Equally clearly, there are exactly as many digits on the diagonal as there are columns. The only reason that the diagonal is longer is that there is sqrt(2) times as much space between each digit and the next because of the way you've drawn it.

You need to stop conflating "completed" with "having a last line". The two notions are not the same.
 
  • #29
jbriggs444 said:
Clearly there exactly as many digits on the diagonal as there are rows. Equally clearly, there are exactly as many digits on the diagonal as there are columns. The only reason that the diagonal is longer is that there is sqrt(2) times as much space between each digit and the next because of the way you've drawn it.

I see that now, that makes sense.

jbriggs444 said:
You need to stop conflating "completed" with "having a last line". The two notions are not the same.

I'm not conflating the two. I understand that completed infinite sets cannot have a last element. I've been working with some numbers I defined that have properties such that they behave as though they did have a last element (in a qualified sense). I'm just seeing if it's possible to duplicate a similar behavior using the existing framework (namely set theory) so that I could derive the existence of these numbers without having to will them into existence.

Incidentally, this thread has been enlightening for me. That "infinite sets exist" is an axiom of set theory seems now to be a boon rather than a real problem. If I'm understanding correctly, I really don't need to worry about trying to "prove Cantor's philosophy wrong" insofar as I am able to define a self-consistent math that ties-in somehow with existing frameworks.
 
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  • #30
hddd123456789 said:
I'm not conflating the two. I understand that completed infinite sets cannot have a last element. .

In the well ordering of the ordinals -which Cantor first defined - [itex]Aleph_{0}[/itex] together with the finite ordinals is an infinite set that has a last element.

[itex]Aleph_{0}[/itex] is the first ordinal that is not finite. Its predecessors in the well ordering are all of the finite ordinals.
 
  • #31
hddd123456789 said:
I see that now, that makes sense.

As an aside, if you compare e.g. the natural numbers and the even numbers there is a very trivial bijection between them, so even though it looks like the natural numbers are "twice as large" they are in fact equally large. So when you make statements like "this set is sqrt(2) times as large as the other set" you have to stop and think about whether that is actually a meaningful and true statement!


I'm not conflating the two. I understand that completed infinite sets cannot have a last element.

This depends on the ordering of the set. Consider the natural numbers with the following ordering: I say that n <' m (<' being my new way of ordering the set) if n > m (> being the normal ordering you are used to on the natural numbers). Then the natural numbers with <' as the ordering has 0 (or 1 depending on who you are :p) as its last element.

In general anytime you want to talk about "last" or "next" or "first" elements, you have to specify how you choose to order a set, and there are a LOT of ways to order sets.
 
  • #32
Right, forgive my ignorance of terminology - again. I just meant to respond to jbrigg's comment about there being a "last line".
 
  • #33
lavinia said:
In the well ordering of the ordinals -which Cantor first defined - [itex]Aleph_{0}[/itex] together with the finite ordinals is an infinite set that has a last element.

[itex]Aleph_{0}[/itex] is the first ordinal that is not finite. Its predecessors in the well ordering are all of the finite ordinals.

I have a very rough idea about these ordinal numbers and from what I read they do seem to have the sort of properties I've been talking about. But as you probably already realized, my grounding in set theory (well math in general) is very weak so am unfortunately not able to add much to this at present. I just happen to have a kind of random interest in infinities and zeroes.
 
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  • #34
hddd123456789 said:
I have a very rough idea about these ordinal numbers and from what I read they do seem to have the sort of properties I've been talking about. But as you probably already realized, my grounding in set theory (well math in general) is very weak so am unfortunately not able to add much to this at present. I just happen to have a kind of random interest in infinities and zeroes.

Cantor was the first to understand the infinite. His theory is what you are looking for.

But one can also find infinite sets with a last element on the number line.

Take the sequence 1/2 1/4 1/8 ... 1/2^n ... This is an increasing sequence that converges to the number 1. It is ordered by size. Put the number 1 in the sequence keeping size as the ordering rule. Then one is at the end of the sequence.

Instead of 1 ,one could also put in any other number larger than 1 as the end of the sequence. I wonder though if Cantor would have given 1 a special place since it is actually defined by the ordering of the powers of 1/2 - that is by the ordering by size. So in some sense, 1 completes the infinity. Nowadays we says that 1 is the limit of the sequence. A set together with all of its limits is called complete.
 
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  • #35
Office_Shredder said:
As an aside, if you compare e.g. the natural numbers and the even numbers there is a very trivial bijection between them, so even though it looks like the natural numbers are "twice as large" they are in fact equally large. So when you make statements like "this set is sqrt(2) times as large as the other set" you have to stop and think about whether that is actually a meaningful and true statement!

This is really more about semantics than the math. I can at least appreciate now that the math, at least the way it's actually written using mathematical notation, is self-consistent. But when you translate it into plain English and say things like "the even numbers are as large as the naturals", I disagree with the philosophy this statement is based on; it requires a rather special definition on what it means for something being "as large as" something else. Though I suppose this likely comes across as obvious to you. And it reinforces me to stop reading popular interpretations of Cantor's work :P
 
<h2>1. What are consecutive reals?</h2><p>Consecutive reals refer to a hypothetical set of real numbers that are arranged in a specific order, with each number being the next consecutive number after the previous one. This concept is often used in mathematics and theoretical physics.</p><h2>2. How are consecutive reals different from regular real numbers?</h2><p>Consecutive reals are different from regular real numbers because they are arranged in a specific order, whereas regular real numbers can be arranged in any order. Additionally, consecutive reals are a theoretical concept and do not have a physical representation, unlike regular real numbers.</p><h2>3. Can consecutive reals exist in the real world?</h2><p>No, consecutive reals are a hypothetical concept and do not have a physical representation in the real world. They are often used in theoretical discussions and mathematical proofs, but they do not have a practical application in the physical world.</p><h2>4. How are consecutive reals used in mathematics?</h2><p>Consecutive reals are often used in mathematical proofs and theoretical discussions, particularly in the fields of calculus and number theory. They can also be used to illustrate concepts such as infinity and irrational numbers.</p><h2>5. Are consecutive reals a proven concept?</h2><p>No, consecutive reals are a hypothetical concept and have not been proven to exist. They are often used as a thought experiment to explore mathematical concepts and theories, but they have not been observed or proven to exist in the real world.</p>

1. What are consecutive reals?

Consecutive reals refer to a hypothetical set of real numbers that are arranged in a specific order, with each number being the next consecutive number after the previous one. This concept is often used in mathematics and theoretical physics.

2. How are consecutive reals different from regular real numbers?

Consecutive reals are different from regular real numbers because they are arranged in a specific order, whereas regular real numbers can be arranged in any order. Additionally, consecutive reals are a theoretical concept and do not have a physical representation, unlike regular real numbers.

3. Can consecutive reals exist in the real world?

No, consecutive reals are a hypothetical concept and do not have a physical representation in the real world. They are often used in theoretical discussions and mathematical proofs, but they do not have a practical application in the physical world.

4. How are consecutive reals used in mathematics?

Consecutive reals are often used in mathematical proofs and theoretical discussions, particularly in the fields of calculus and number theory. They can also be used to illustrate concepts such as infinity and irrational numbers.

5. Are consecutive reals a proven concept?

No, consecutive reals are a hypothetical concept and have not been proven to exist. They are often used as a thought experiment to explore mathematical concepts and theories, but they have not been observed or proven to exist in the real world.

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