It is basically the main point of the definition of ordinals that for any property of ordinals , there is a first ordinal with that property. There are, however, foundational theories without uncountable ordinals , for instance Nik Weaver's Mathematical Conceptualism.

Well, that depends on what you take to be decent. In the sibling, shinoteki has pointed (via Nik Weaver) to J_2. As Weaver argues, this is plenty strong enough to do ordinary mathematics: the mathematics that most mathematicians work on, and the mathematics that (almost always, perhaps absolutely always) is used in real-world applications. On the other hand, I find it difficult to work with, and prefer explicit reasoning about sets (but I'm a mathematician, so maybe I'm just used to that). That said, I think that properly limiting the impredicativity of set-based constructions should allow one to create a set-like theory that corresponds to something like J_2. (I'm being vague here because I don't know better; it's possible, I'd even say likely, that other mathematicians know better responses.)

That depends on what you mean by "well-ordered". My philosophy of doing constructive mathematics (mathematics without excluded middle, and often with other restrictions) is that one should define terms as much as possible so that the usual theorems (including the theorems that the motivating examples are examples) become true, so long as the definitions are classically (that is using the usually accepted axioms) equivalent to the usual definitions.

As the motivating example of a well-ordered set is the set of natural numbers, we should use a definition that makes this an example. Such a definition may be found at a math wiki where I contribute my research (such as it is). Then (adopting a parallel definition of "ordinal") it remains a theorem that every set of ordinals is well-ordered.

What about something like 10^100, i.e., something you could easily wright out in decimal but couldn't count to?

Why stop at big numbers? Even the numbers you handle in everyday life might lead to a false statement, you are not logically omniscient and therefore wouldn't necessarily know if they did. Why not be uncomfortable with everything?

I wonder how much of this is just a function of what math you've ended up working with a lot.

Humans have really bad intuition about math. This shouldn't be that surprising. We evolved in a context where selection pressure was on finding mates and not getting eaten by large cats.

Speaking from personal experience as a mathematician (ok a grad student but close enough for this purpose) it isn't that uncommon for when I encounter a new construction that has some counterintuitive property to look at it and go "huh? Really?" and not feel like it works. But after working with the object for a while it becomes more concrete and more reasonable. This is because I've internalized the experience and changed my intuition accordingly.

There are a lot of very basic facts that don't involve infinite sets that are just incredibly weird. One of my favorite examples are non-transitive dice. We define a "die" to be a finite list of real numbers. To role a dice we pick a die a random number from the list, giving each option equal probability. This is a pretty good representation of what we mean by a dice in an intuitive set. Now, we say a die A beats a die B if more than half the time die A rolls a higher number than die B. Theorem: There exist three 6-sided dice A, B and C with positive integer sides such that A beats B, B beats C and C beats A. Constructing a set of these is a fun exercise. If this claim seems reasonable to you at first hearing then you either have a really good intuition for probability or you have terrible hindsight bias. This is an extremely finite, weird statement.

And I can give even weirder examples including an even more counterintuitive analog involving coin flips.

I just don't see "my intuition isn't happy with this result" to be a good argument against a theorem. All the axioms of ZF seem reasonable and I can get the existence of uncomputable ordinals from much weaker systems. So if there's a non-intuitive aspect here, that's a reason to update my intuition not to reduce my confidence in set theory.

ETA: If you want to learn more about this (and see solution sets for the three dice problem) see this shamelessly self-promoting link to my own blog or this more detailed and better written Wikipedia article.

What does "existence" have to do with anything though? Even if the real world, or morality don't "exist" in some sense, you still go on making decisions, reason about their properties. (There appear to be two useful senses of "doesn't exist": the state of some system is such that some property isn't present; or a description of a system is contradictory. These don't obviously apply here.)

The trouble is, human value might turn out to talk about complicated mathematical objects, just like mathematicians can think about (simpler kinds of) them, and it's not clear where to draw the line, at least to me while I still have too little understanding of what humane value is.

Reasoning about math objects, as opposed to just patterns of the world, seems to be analogous to reasoning about the world, as opposed to just patterns in observation. I don't believe there has to be a boundary around physics, a kind of "physical solipsism". And limitations of representation don't seem to solve the problem, as formal systems might be unable to capture particular classes of models of interest, but they can be seen as intended to elucidate properties of those models.

the Burali-Forti paradox saying that the predicate "well-ordered" cannot be self-applicable ... I just have trouble believing that there's actually any such thing as an uncountable ordinal out there, because it implies an absolute well-ordering of all the countable well-orderings; it seems to have a superlogical character to it.

I don't think that it's fair to characterise the B-F paradox this way. The argument of B-F is that, given any collection S of well-orderings closed under taking sub-well-orderings, S cannot be among the well-orderings represented in S itself. There is nothing paradoxical here. (I'm not sure whether this matches the content of Cesare Burali-Forti's 1897 paper, which I haven't read and of which I've heard conflicting accounts, but the secondary sources all seem to agree that he did not believe that he had found a paradox. ETA: After following the helpful link from komponisto, I see that sadly this is not how B-F himself viewed the matter.)

Now, if you add the assumption of an absolute collection of all well-orderings, then you get a paradox. But an absolute collection of (say) all finite well-orderings leads to no paradox; we just know that this collection is not finite. And an absolute collection of all countable well-orderings leads to no paradox either; we just know that this collection is not countable. And so on.

Of course, none of this shows that such collections actually exist. If you said that you don't really believe in uncountable ordinals (perhaps on the grounds that they're not needed for applications of mathematics to the real world), I would not have commented (except maybe to agree); but calling them incredible (as you seem to do, counting them as evidence against set theory, indeed among the strongest that you know) goes far beyond what I would consider justified.

There seems to be controversy about "exist" and "out there", can you taboo those?

For example, are you saying the you think Ultimate Ensemble does not contain structures that depend on them, or that they lead to an inconsistency somewhere, or simply that your utility function does not speak about things that require them, or what exactly?

The following is a comment by John Baez, posted on Google+ where I linked to this thread:

It's indeed hard to believe, at a gut level, in the existence of a well-ordered uncountable set. For example: can you take the set of real numbers and linearly order them in some funny way such that any decreasing sequence of them, say a > b > c > ..., "bottoms out" after finitely many steps? (Here > is defined in the funny way you've chosen.) Nobody knows an explicit way to do this, and you can prove that nobody ever will. Yet the "well-ordering theorem" says you can do it:

http://en.wikipedia.org/wiki/Well-ordering_theorem

What's the catch? This theorem is equivalent to the Axiom of Choice, which cannot be proved (or disproved!) from the rest of the Zermelo-Fraenkel axioms of set theory.

So, we may decide to disbelieve in the Axiom of Choice. But there are other ways of stating it, which make it sound obviously true.

Set theory is just a made up bunch of puzzle pieces (axioms) and some rules on how to fit them together (logic) so it's weird to hear you lot talking about "existence" of a set with some property P as something other than whether or not the statement "exists X, P(X)" has a proof or not. I thought Hilbert's finitist approach should have slain Platonism long ago.

...I just have trouble believing that there's actually any such thing as an uncountable ordinal out there, because it implies an absolute well-ordering of all the countable well-orderings; it seems to have a superlogical character to it.

I thought that could be proven without reference to the existence of a set of them, just from general facts about well-ordering? And then the only question is whether the class of all countable ordinals is set-sized. Which it must be since they can all be realized on N. As long as you accept the continuum, anyway! I don't see how the continuum can possibly be more acceptable than omega_1.

For essentially the same reasons I have trouble believing that the first infinite ordinal exists.

Finite ordinals are computable, but otherwise your remarks still apply if you swap out "countable" for "finite." According to ZF there are uncomputable sets of finite ordinals, so you can't verify that they are well-ordered algorithmically.