Consecutive Reals: A Hypothetical Possibility?

[Office_Shredder]

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.

 

[hddd123456789]

Right, forgive my ignorance of terminology - again. I just meant to respond to jbrigg's comment about there being a "last line".

 

[hddd123456789]

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

Aleph_{0} 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.

 

[lavinia]

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.

 

[hddd123456789]

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

 

[jgens]

This is really more about semantics than the math.

To an extent yes. Although the mathematics informs the definitions. With only set level data, there are not very many useful ways of talking about the relative sizes of sets. Cardinality is crude but it does manage to capture some useful information.

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.

Not really in my opinion. Consider the set {a,aa,aaa,...} consisting of all finite strings of the letter a. Most people would agree this set is equally as large as the natural numbers. That the set of even natural numbers is equally as large as the natural numbers is an immediate corollary. We just change the labels!

Edit: The real problem is not that this definition of size is somehow incongruous with the way people naturally think about things, but rather that most people have inconsistent notions of size that they try applying simultaneously.

 

[hddd123456789]

Not really in my opinion. Consider the set {a,aa,aaa,...} consisting of all finite strings of the letter a. Most people would agree this set is equally as large as the natural numbers. That the set of even natural numbers is equally as large as the natural numbers is an immediate corollary. We just change the labels!

I'm not most people in that case. Again that requires that one accept that there is such a thing as a set of all naturals. Apparently we can see the logical consequences of such a set anyway, but I personally find it akin to asking the unqualified question "how many apples are there?". As many as one likes I say. It just requires somehow consistently and usefully defining unique numbers larger than any of the type (1+1+1+...).

 

[jgens]

Again that requires that one accept that there is such a thing as a set of all naturals.

True. But then claiming the statement that even numbers are equinumerous with the natural numbers is somehow jarring to your intuitive definition of size is kind of silly. You have no belief that such things exist anyway!

In any case, there are lots of reasons to use infinite sets. The first is utility and the widespread use infinite sets have found describing the natural world. The second is aesthetics in that doing math without infinite sets, while possible, gets quite ugly. The third is that even talking about problems in elementary arithmetic gets quite difficult if you discard the notion of infinite sets. For example, one common proof technique works by establishing that a proposition is true for the base case and then showing that if it holds for some arbitrary case, then it must also hold for the next one. Unfortunately without infinite sets you cannot quantify the statement over the set of all natural numbers so this natural proof technique gets broken. One way to fix this involves turning ordinary theorems in arithmetic into metatheorems, much in the same way proper classes are dealt with in ZFC, but this is less than satisfying in my opinion.

Apparently we can see the logical consequences of such a set anyway

Of course we can. The same criticisms can be applied to any axiom of set theory. One could object that the empty set does not really exist or that sets are not actually well-founded. The whole of axiomatic set theory is about deriving the logical consequences of such sets.

 

[hddd123456789]

True. But then claiming the statement that even numbers are equinumerous with the natural numbers is somehow jarring to your intuitive definition of size is kind of silly. You have no belief that such things exist anyway!

That's debatable really, but I won't go there.

In any case, there are lots of reasons to use infinite sets. The first is utility and the widespread use infinite sets have found describing the natural world. The second is aesthetics in that doing math without infinite sets, while possible, gets quite ugly. The third is that even talking about problems in elementary arithmetic gets quite difficult if you discard the notion of infinite sets. For example, one common proof technique works by establishing that a proposition is true for the base case and then showing that if it holds for some arbitrary case, then it must also hold for the next one. Unfortunately without infinite sets you cannot quantify the statement over the set of all natural numbers so this natural proof technique gets broken. One way to fix this involves turning ordinary theorems in arithmetic into metatheorems, much in the same way proper classes are dealt with in ZFC, but this is less than satisfying in my opinion.

By the way, I have no issue with infinite sets. I just take issue with Cantor's treatment of them. I don't see them as particularly useful. One would think if they were useful, I would at least have heard of his name in my analysis course in college, let alone his cardinals or ordinals. Instead I was taught strictly limits and that infinity is not a number. Granted it was Calc 1 but you'd think his work would at least be mentioned in passing.

 

[jgens]

I just take issue with Cantor's treatment of them.

The ordinals and cardinals are actually more benign than the infinities used in analysis. Lots of arguments in analysis utilize choice principles, which require further assumptions in addition to those utilized by Cantor.

I don't see them as particularly useful.

This just indicates you have seen very little mathematics. Not that ordinals and cardinals are useless. If you want to see their utility, then the simple solution is to study more mathematics.

One would think if they were useful, I would at least have heard of his name in my analysis course in college, let alone his cardinals or ordinals. Instead I was taught strictly limits and that infinity is not a number. Granted it was Calc 1.

Well introductory calculus and analysis is not really the place where ordinals/cardinals would arise anyway. In truth the ordinal and cardinal numbers are behind some very deep theorems in mathematics and are often used implicitly in other cases. Lastly the infinities in calculus and analysis are very different than the infinites in the theory of transfinite numbers. You seem to be confusing the two which is something that should stop.