Re: arithmetic in ZF

Hi Graham, Dr. Priest,

On sci.logic we are discussing arithmetic in ZF and got to discussion
of the Domain Principle. Your book "Beyond the Limits of Thought" was
quoted, I wonder if you might have something to say, on sci.logic on
the thread "arithmetic in ZF".

http://groups-beta.google.com/group/sci.logic

Where I'm coming from, I'm an amateur logician who advocates a theory
free of non-logical axioms, and think that that theory can thus be
Goedelianly complete, and the axioms of set structure of ZFC minus the
regularity axiom are theorems of the Null Axiom or Axiom-Free theory,
which is a theory with sets, numbers, or physical or geometric objects
as primary objects, at once.

Basically it has an ur-element that is dually minimal and maximal, I've
gotten to calling it "dually-self-intraconsistent". In scanning some
few words of yours written on the Internet, and about dialetheism, it's
basically about Janus' introspection. I have the singular ur-element,
which is as well a set and the proper class, where there can be only
one proper class, being at once the root of the Liar, the Russell set,
infinity, the empty set, Kant's Ding-an-Sich and Hegel's Being and
Nothing, and the void from which all springs. Particularly for the
Liar and Russell, and parallelly for Cantor/Burali-Forti/apeiron, I've
discussed that on sci.math and sci.logic for some years.

I think infinite sets are equivalent, I show that, basically with
ubiquitous ordinals or naturals in the cumulative hierarchy, and work
on some analytical tools that have to do with bijections between the
natural integers and unit interval of the real numbers, with
nonstandard real numbers that have atomic infinitesimal iota-values,
indubitably, as a logical consequence of their structural consequence,
for the normal ordering of the positive reals being its natural
well-ordering.

I don't only promote a theory with zero non-logical (or proper) axioms,
I promote that it's first-order logic and that it's the only true
theory.

This is basically towards Deep Foundations, and to some extent a theory
of everything.

I hope this serves as a decent introduction, my name's Ross, Ross A.
Finlayson, USA, I post this to you via e-mail and onto that thread on
the sci.logic discussion forum. I'm interested in your opinion, and
would be made happy to receive a reply, publicly or privately.

Thank you,

Ross Finlayson

.

Relevant Pages

  • Re: Skolems Paradox and why is math the way it is?
    ... the class of singleton subset of the reals is such a big class. ... >>of the naturals, but I've seen no proofthat this set has no logical ... that is a FAITHFUL model of the ZF axioms. ... two singleton subsets of the reals and give it to the person in EACH ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... The same axioms that give us transfinite set theory are needed to ... give us the naturals, and from them the other number systems on ... The invertible functions of IFR map reals to reals. ... infinite number. ...
    (sci.math)
  • Re: Why?
    ... cardinality of the natural numbers. ... naturals to that set) and being subcountable: ... reals is greater that the cardinality of the set of natural numbers. ... from whatever axioms are active in the proof of the theorem. ...
    (sci.logic)
  • Re: infinity
    ... exists real positive non-zero, say, iota, less than any other positive ... In a circular form of argument, in well-ordering the reals there is a ... The natural naturals, are, how you say, as from Peano characterization, ... Sounds second-order to me, Ross. ...
    (sci.math)
  • Re: An uncountable countable set
    ... "Finlayson Numbers" known to anyone except Ross himself? ... My understanding, looking at each of these axioms, is that they apply to ... reals are defined as the immdiate neighbors, ...
    (sci.math)