Re: Countable models of ZFC

On Oct 5, 3:26 pm, "Nam D. Nguyen" <namducngu...@xxxxxxx> wrote:
Rupert wrote:
As far as philosophy goes: do you understand the first-order language
of set theory or don't you?

What does understanding first-order language of set theory have to
do with philosophy here?


I was under the impression that we were discussing philosophical
concerns arising from the question of how we interpret sentences in
the first-order language of set theory. George was talking about which
model my definition should be interpreted in, for example.

If you don't, then I guess you're going to
have a problem understanding mathematics.

Must one understand the first-order language of set theory, to have no
problem in understanding, say, the mathematics of the naturals?


No, of course not. It is quite coherent to say that you understand the
first-order language of arithmetic, but not the first-order language
of set theory. Such a position is perfectly reasonable.

But most mathematicians
think they can understand the first-order language of set theory.

I suppose so.

They feel that they know what they're talking about.

I think a lot of people feel that way too, whether or not they
actually realize what they're really ... really talking about!

They feel that they
can make sense of the idea of a sentence of the first-order language
of set theory being true, not just true in this model or that model of
the axioms.

So I suppose 50% of them feel AC is true, and the other false! What kind
of mathematics would that be then?


Nobody worries too much about AC these days. In the early days there
was significant controversy. These days we accept ZFC as the "gold
standard" of mathematical rigour. Some people prefer weaker systems,
others stronger, but when we are publishing mainstream mathematics it
is assumed that ZFC is the background theory unless explicitly stated
otherwise.

Are you raising worries about the epistemology of mathematics?


If you don't feel this idea has any meaning, fine, but
then this will be a problem with all of mathematics, not just the part
I'm talking about.

What are you talking about here?


I made a perfectly ordinary mathematical statement and George starting
ranting and raving about it. There really wasn't any cause to. It was
no more problematical than any other mathematical statement. There
were no special philosophical issues raised by it that are not equally
raised by any other mathematical statement.



--
hz


.

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