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Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V32 (Analytical knowledge)

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olcott <NoOne@NoWhere.com> writes:
[...]
> The semantic meaning of the conceptual class of natural numbers is
> defined as an ordered set of names that are associated with an ordered
> set of finite strings of numeric digits. Functions and predicates have
> also been stipulated to define semantic meaning this conceptual class.
[...]

Why would you define the natural numbers that way?  Sequences of
digits (I'm guessing you meant decimal, but you didn't say)
are just a convenient way of representing them -- and it doesn't
make sense until you've defined or derived a lot more knowledge.
You need addition and multiplication before you can even say what
"123" means, or demonstrate that 9+1=10.

If you're trying to define them from first principles, you can
start by saying that 0 is a natural number and the successor of a
natural number is a natural number.  You'll need a few more axioms.
See Robinson Arithmetic, for example.  You'll recall the "0", "S0",
"SS0", ... notation we've discussed here before.

If you insist on defining natural numbers as "finite strings of
numeric digits", does that include the empty string?  Are "1",
"01", and "001" the same natural number?  If so, what reasoning
lets you conclude that they are?

-- 
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
Working, but not speaking, for Philips Healthcare
void Void(void) { Void(); } /* The recursive call of the void */

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Simply defining Gödel Incompleteness and Tarski Undefinability away V32 (Analytical knowledge) olcott <NoOne@NoWhere.com> - 2020-07-23 15:01 -0500
  Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V32 (Analytical knowledge) André G. Isaak <agisaak@gm.invalid> - 2020-07-23 14:49 -0600
  Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V32 (Analytical knowledge) Keith Thompson <Keith.S.Thompson+u@gmail.com> - 2020-07-23 15:08 -0700
    Re: Simply defining Gödel Incompleteness and Tarski Undefinability away V32 (Analytical knowledge) André G. Isaak <agisaak@gm.invalid> - 2020-07-23 16:51 -0600

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