Re: Post Axiom Syndrome

It seems to me that lots of people quite interested in mathematical
logic, for their own sake or that of others, are basically milling
about either looking for something they find better than other modern
foundational perspectives, or for a smaller group, generally comprised
of academicians, promoting their own notions as to what should be used
to describe the foundations of mathematical logic.

They seek an alternative because they find the status quo unsatisfying,
for a variety of reasons. Consider the Liar paradox. With primary
objectives of consistency, that no untrue statements are provable, and
completeness, that all true statements are provable, a statement as the
liar is basically false, it's not true, but then of course its
statement is that its false, so its not false, but not true, false in a
different or circular sense.

The Liar is not so bad, but with further primary objectives in an
encompassing theory of mathematical logic, unrestricted comprehension
and quantifiers lead to a variety of states or modes of statement that
that basically reflect the Liar. These "paradoxes" are generally or
can be referred to be the poor fellow for whom it is named, for example
Cantor, Russell, Burali-Forti. Some statements of paradoxical
situations are somewhat between the set-theoretical, with associated
structure, and plain Liar, by itself, for example Grelling,
self-referential invalidating circularity.

Then, above the structure generated by some of the basic tools of
Boolean and first order predicate logic, there is basically the theory
of numbers, where that structure might be seen as that of the most
primitive aspects of individuation and uniquification, ie after
describing a primary object or ur-element, the numbers can readily
follow without much intervening structure. Then, in numbers, there are
generally those notions of infinitesimals vis-a-vis finite and even
infinite numbers, or basically those issues named for Zeno of Elea, or
as well Cantor/Burali-Forti, where what is called Cantor's paradox and
what is called Burali-Forti's paradox can be seen to share several
aspects, and the paradoxes of Zeno find themselves to be about
basically induction.

So anyways, if you actually care what the very notion of "truth" means,
and care to be able to prove succinctly to any other a mathematical
fact, you might find some utility in axiom-free or axiomless systems.

ZF is inconsistent, because the set-theoretical universe is infinite,
among other reasons that resolve to the same thing.

Ross

.

Relevant Pages

  • Re: My investigations into Godels Incompleteness Theorem
    ... the Liar, is not literally self-referential ... understood or explained in the literature. ... When CBL is used to formalize the Liar Paradox, ... particular sets and systems of representation we formalize these ...
    (sci.logic)
  • Re: Solution to Liar Paradox
    ... It is like Russell's paradox: ... why is the Liar not such sentence in spite of all appearances? ... The Liar simply foregos completeness, ... self-reference, negation, substitution and consistency. ...
    (sci.logic)
  • Re: Solution to Liar Paradox
    ... English sentences must have truth assignments that satisfy them. ... Having an allowable truth assignment does not by itself solve the Liar ... For example, consider Curry's paradox; ...
    (sci.logic)
  • Re: Why Regularity?
    ... that no language can define its own truth predicate. ... There's a more recent treatment of the paradox in "The Liar", ... Is my analogy between the liar paradox and ZFC consistent? ...
    (sci.math)
  • Re: Galileos Paradox and the Project of the Reals
    ... Galileo Paradox of about a month ago. ... When one compares the (infinite) sequence of squares with the ... sequence of naturals, there seems, on the one hand, to be more naturals ... the table of reals is square then the diagonal argument doesn't work. ...
    (sci.math)
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