Re: Looking for Undecidable Propositions in Systems without a certain amount of arthimetic.

On Aug 12, 2:01 pm, Scott <ToaTe...@xxxxxxxxx> wrote:
Hi: In order to apply Godel's incompleteness theorem, we need "a
certain amount of arithmetic" in the formal system, which can be
loosely translated to natural numbers and a couple of basic rules
about addition and multiplication. Does anyone know of undecidable
propositions in formal systems where natural numbers (or equivalent)
are not present; ie, the formal system does not meet the requirement
of "a certain amount of arithmetic"?

The pure predicate calculus in any given language is one. The pure
predicate calculus in any given language is the least theory in that
langauge. Then any contingent sentence in that language is undecidable
in the theory.

Of, for a theory with at least one non-logical axiom:

The only non-logical symbol is 'F', a 2-place predicate symbol (so
this is essentially the language of set theory). The only axiom is:

AxEy~Fxy

An example of an undecidable sentence in this theory:

AxFxx

MoeBlee

.

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