Re: Would it matter if ZF was inconsistent?

On Jul 22, 4:02 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jul 22, 3:56 pm, "Ross A. Finlayson" <ross.finlay...@xxxxxxxxx>
wrote:

ZF's universe is the Russell set, so it
contains itself.

You keep saying that no matter how many times it's been explained to
you why you're incorrect, incoherent even. You actually thrive on
remaining ignorant of the basics of the subject, ignoring what has
been explained to, and instead remumbling various falsehoods and
nonsense over and over in ad hoc variations.

MoeBlee

That short proof uses several features of modern, standard
mathematics.

It's kind of like different interpretations of the use of model theory
and the description of the halting problem. In model theory, there's
always a bigger ordinal to contain structures of interest, so it is
said that there is a model of each construct of this regular set
theory (completeness). In the halting problem, it's said that there's
always a bigger input than an algorithm can handle, so there isn't.
Conversely, there's always a bigger structure (eg the model) to model,
and always another algorithm than can handle that input.

Here, ZF is incomplete so there's always a bigger theory, and
maintaining ZF's axioms in that supertheory, a set theory, ZF's
universe among, within, and as part of the other is too big not to
contain itself.

Why would I care whether or not ZF was inconsistent? Partially it's
because there are classical constructions (with application) that see
better treatments in currently nonstandard methods, and some that
aren't consistent with standard methods. Besides the classically
anecdotal like the projectively extended real numbers and their plane
(with its applications), there are also notions like the equivalency
function (EF is a CDF, of a distribution even built from standard
ordinals exploiting symmetry). As well, nobody uses transfinite
cardinals in physics, but there are lots of things to do directly with
infinities and infinitesimals that are more directly considered with
regards to currently nonstandard mathematical constructions not
consistent with ZF.

Apply nested intervals to a well-ordering of the reals, there aren't
degenerate intervals except from adjacent points. The reals are
consistently equivalent to a wide variety of cardinals, yet in that
way to their initial ordinals, which exist or don't, and only to one
of those. The infinite binary tree's paths are dense among countable
points dense among them. These are more examples leading to
considerations that ZF is inconsistent.

ZF is incomplete (or inconsistent). Of course, finite combinatorics
of finite structures can be perfectly modeled using a subset of the
axioms of ZF. Now, where in assuming ZF consistent, it's incomplete,
would you ever expect to find something "true", and useful, about
mathematical structures of interest, beyond it? If so, in
application, in analysis, then it would be in mathematical primitives
that offer utilitarian features to explain the reality of the big and
small. Transfinite cardinals (and extensions to Nonstandard
infinitesimals that aren't) haven't been found to accomplish that.

So, I disagree and dispute that, Moe.

Ross
.

Relevant Pages

  • Re: parallels
    ... I strongly suggest you replace your consistent rhetoric (previously I ... If you reverse them then you reverse their ... indefinitely becoming amazingly complex (as they are within mathematics). ... provide just a single axiom which is not a tautology. ...
    (sci.bio.evolution)
  • Re: Complex numbers in the plane, axiom or consequence?
    ... David C. Ullrich ha escrit: ... Actually saying that there's no way to show PA is consistent ... as a basis for mathematics are assumed to be consistent. ... with the other axioms". ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... I mean, as a physicist, I have no problems with infinitesimals .. ... None of these things is forbidden in mathematics. ... Which is a capsule description of TO's operating principles. ... their decrees as to truths which they base on unfounded axioms. ...
    (sci.math)
  • Re: When is a logic a quantum logic?
    ... > mathematics we seek the simplest consistent axiom set, ... > many valued logic is a consistent mathematical structure by reducing it ... Boolean logic some of the expressive power of topological spaces. ... The "non-classical" logic that Sikorski refers to are modal logics, ...
    (sci.physics.research)
  • Re: infinity
    ... the most "real" of the infinite systems, as it is the one that is most ... widely used across different areas of mathematics. ... ideal point at infinity to give the real line topological equivalence ... for any arguments which support them, consistent or not). ...
    (sci.math)