Re: Godel cant tell us what makes a mathematical statement true



Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
herbzet wrote:
You are arguing against the practice of giving intuitive,
unformalizable arguments as proof, a practice which no one
is advocating.
Your statement could be interpreted in 2 ways.

First way: The "practice of giving intuitive unformalizable
arguments as proof" is a practice "which no one
"is advocating". And I'm arguing against such a
dismissed practice. So I must be doing something
right, correct?

The above is the intended interpretation.

Unfortunately, that's how we've done reasoning after Hilbert's
Program. I'm not defending Hilbert's one-size-catch-all formal
system, only his syntactical provability. What we've done since
Godel is to replace that formal system with one-encoding-size-catch-all
arithmetic (interpretation) truth system.

I'm not sure what you're talking about with this last sentence.

Could you expand a bit on what you mean by "What we've done
since Godel is to replace that formal system with one-encoding-
size-catch-all arithmetic (interpretation) truth system"?

--
hz








Talking about avoid one "devil", just to meet another one!


Second way: My "arguing against the practice..." is something
"no one is advocating". So I'm doing something not
popular here. But why should "intuitive, unformalizable
arguments" be accepted as proof, while *proof* has a
very rigor definition that everyone already accepts?

--
hz

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

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