Re: Peano's Axioms are Inconsistent

Joshua Cranmer wrote:
Joshua Cranmer wrote:
I proved that AxEy T(y)(x) = '1'. Now, if you want to show that PA is inconsistent, you have to show that it assumes that the negation of that proposition is also true, namely ExAy T(y)(x) != '1'. It's your turn to prove it.

I forgot to mention: if you add any more axioms as assumptions, all you've done is proven that the system with those axioms is inconsistent. So if you want to prove that PA is inconsistent, you must only assume the axioms of PA, no more.


One of my hobbies is trying to identify unstated added axioms in
attempts to prove inconsistency of well studied systems, such as PA.
This thread gave me some trouble, but I think I have identified the
added axiom:

=======================================================================
If P and Q are two propositions, and there exists an English sentence E
such that P and Q are both possible expressions of E in mathematical
notation, then P and Q are equivalent.
=======================================================================

I am going to call this the "Axiom of Ambiguity", because E can be as
ambiguous as English permits. The nature of this axiom explains
Russell's failure to express his proof in formal terms. Any proof that
applies it contains at least one English sentence

I believe the axiom of ambiguity is being applied in this thread with

E = "The BNTM writes to every cell" (or some variation on it)

P = AxEy T(y)(x) = '1'
Q = EyAx T(y)(x) = '1'

The Axiom of Ambiguity explains the otherwise mysterious jump from P to Q.

It is the most powerful and general purpose added axiom I have ever
seen. It is quite easy to construct ambiguous English sentences,
especially when discussing infinite sequences and the like. I suspect
that adding the Axiom of Ambiguity to any formal system would result in
an inconsistent system.

Unfortunately, the axiom itself cannot be formalized - the notion of
what propositions might be considered valid expressions of an ambiguous
sentence is too dependent on natural language interpretation. In the
example above, it is highly arguable whether Q really is a valid
expression of E.

As you point out above, proving inconsistency of PA + Axiom of
Ambiguity, or any other added axiom, is no help at all with proving
inconsistency of PA.

Patricia
.

Relevant Pages

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