Re: FOL and Infinity
 From: sudhir <sudhir_sh@xxxxxxxxxxx>
 Date: Tue, 12 Apr 2011 11:10:15 0700 (PDT)
On Apr 12, 12:39 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Apr 11, 1:27 pm, sudhir <sudhir...@xxxxxxxxxxx> wrote:..
This is precisely what I said. Your definition of AxPhi(x) is "For anyHow 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.
constant c
in the language, AxPhi(x) implies Phi(c)'.
You cannot explain or define AxPhi(x) without using 'For any(all)'
That is where you are into
an interminable regress.
The sentence Phi(1) &Phi(2) implies each of Phi(1) and Phi(2).
The sentence Phi(1) &Phi(2) &Phi(3) implies each of Phi(1) , Phi(20
and Phi(3).
One could go on like this, but there is no sentence of sentential
logic which will imply
Phi(i) for every i.
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.
The big assumption is that such a limit exists; 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. AxPhi(x) is a shorthand for
an infinite amount of information ; it is the first idol in the Temple
of Infinity.
..
.
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