Re: what makes it true?


>> Like I said, I can follow the rules of the system without having a set
>> theory. However, to prove anything about the system, such as
>> consistency, I have to be able to talk about the set of statements
>> in the theory, so I need a set theory. To be able to talk about the set
>> of provable statements, I'll need to actually talk about sets. To talk
>> about truth in the theory, I need to talk about models of the theory.
>> Since a model is a set, that requires a set theory. To ask whether
>> a statement is independent from the theory, I need to be able to talk
>> about whether it is in the set of provable statements from the theory.
>> I don't see any way around it.

> I see, and this is how you learned to understand "and so on"?

I learned to *understand* 'and so on' by proving things by induction.
That is very different from the *intuition* that I picked up as
a kid. But I've also learned that my intuitions can be very unreliable,
even inconsistent at times, which is why I want things proven. The
phrase 'and so on' is one of those notorious ones that tends to defy
intuitions.

As Little pointed out, we obtain intuitions about the natural numbers
as kids, but it isn't at all clear that your intuition and my intuition
are the same. Adopting an axiom system is what guarantees that we can speak
the same language and agree on at least some propositions. But for an
independent statement we have to either adopt another axiom which
decides the matter, agree to disagree, or give up trying to make a
decision. Your intuition and mine may very well disagree about new axioms,
although since I adopt at least ZFC and consider the natural numbers
as a set in that theory, I have a feeling we will agree on quite a bit.

But, for example, I tend to like the Generalized Continuum Hypothesis.
Many logicians tend not to like it. For me, the unity that it gives to
cardinal arithmetic overides the vague ideas about cumulative heierarchies.
I also have a tendency to dislike measurable cardinals because they lead
to extra hypotheses in theorems on measre theory. This dislike isn't
strong, and it may be possible to convince me of their utility if
some amazing theorems can be proved using them. But the decision is
made at the level of intuition and pragmatism, bot at the level of
'truth'.

The upshot is that when you define the naturals as 0,1,2 and so on, the
difficulty is exactly in the phrase 'and so on'. It is way, way too
vague to be in a definition.


--Dan Grubb
.

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