Re: FOL and Infinity

On Apr 12, 1:10 pm, sudhir <sudhir...@xxxxxxxxxxx> wrote:
On Apr 12, 12:39 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:> On Apr 11, 1:27 pm, sudhir <sudhir...@xxxxxxxxxxx> wrote:
How does the sentence AxPhi(x) imply Phi(1) ,
unless some intuitive semantic content
is read into AxPhi(x)?

No matter what the interpretation of the language, AxPhi(x) implies
Phi(c) for any constant 'c' in the language. That is, for any constant
'c' in the language, there is no model in which AxPhi(x) is true but
Phi(c) is false.

This is precisely what I said.

No, it's not.

Your definition of AxPhi(x) is "For any
constant c
 in the language, AxPhi(x) implies Phi(c)'.

No, I didn't give a "definition" of a formula AxPhi(x). Indeed, I said
that I don't know what you mean by a "definition" of such a formula.
You skipped what I wrote.

You cannot explain or define AxPhi(x) without using 'For any(all)'
That is where you are into
an interminable regress.

I already addressed this in a previous post. You skipped what I wrote.

there is no sentence of sentential
logic which will imply
Phi(i) for every i.

Phi(i) is not even a formula of sentential logic.

Precisely such a sentence is introduced  in FOL , namely  AxPhi(x) .In
a way, AxPhi(x) is
a limit to the sequence
Phi(1), Phi(1)&Phi(2), Phi(1)&Phi(2)&Phi(3), and so on.

You like to use words like "limit" in your own personal way, it seems.

The big assumption is that such a limit exists;

No, there's no notion of such a limit when we formulate a language so
that "AxPhi(x)" is a formula of the language.

You're just making stuff up.

that such a sentence
(comprising of a finite string of symbols)
can convey that each of the independent sentences Phi(1),Phi(2) and so
on is true.

Why don't you just read a book where you'll find the inductive
definition of "true in an interpretation"?

AxPhi(x) is a shorthand for
an infinite amount of information ; it is the first idol in the Temple
of Infinity.

Right, by saying "idol" and "Temple" you really convinced me that you
have a cogent argument even though you don't know Jacques about
anything here.