Consecutive Reals: A Hypothetical Possibility?

[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.