Re: ZFC in another shape.

On Feb 5, 2:33 pm, "zuhair" <zaljo...@xxxxxxxxx> wrote:

Primitives: e,=

I suppose your theory is an extension of first order logic with
identity? I asked that question before about other of your "theories".

Definition Schema 1 )

x is an ordinal <-> AyAz((yex & zey)->zex)&Ay(yex->AzAw((zey & wez) ->
wey))).

The number of left parentheses do not match the number right
parentheses. You probably mean:

x is an ordinal <-> (Ayz((yex & zey) -> zex) & Ay(yex -> Azw((zey &
wez) -> wey)))

And you don't need to call that a 'schema'. Just call it a
'definition'.

Axioms:-

1) Extensionality
2) Foundation
3) Ordinal Comprehension:

AdExAy(yex<->((yed \/ y=d) & d is an ordinal )).

At a quick glance, this seems to assert that for any ordinal d, we
have the existence of the successor of d. Why not just call it the
'ordinal successor axiom'?

4) Replacement

Which formulation of replacement schema? (And, by the way, replacement
is a schema, not a just a single axiom). I found out in a discussion
that it may make a difference which particular formulation of
replacement is used along with the inclusion or exclusion of certain
other axioms.

5) Infinity

You have not defined '0', 'u' and '{}' that are used in the axiom of
infinity. If you don't first prove the needed existence theorems, then
your axiom might not say what you want it to say. I already pointed
this out before.

Or, just state your axiom of infinity in primitive notation.

6) Choice

You have not defined 'is a function' nor functional notation that are
usually used for this axiom. If you don't first prove the needed
existence theorems (for unordered pairs, upon which the definition of
'is a function' ultimately relies for 'is a function' to have any more
than a "hypothetical force") then your axiom might not say what you
want it to say. I already pointed this out before.

Or, just state your axiom of choice in primitive notation.

7) Power.

It seems that the existence of the empty set will depend on your
particular formulation of the axiom schema of replacement.

I have no idea how to prove the union axiom from your axioms.

MoeBlee

.

Relevant Pages

  • Re: My last theory.
    ... Let us see how this Replacement schema works. ... in Axiom Schema 8. ... Schema only proves the existence of the images of functions -- ...
    (sci.logic)
  • Re: My last theory.
    ... Let us see how this Replacement schema works. ... in Axiom Schema 8. ... Schema only proves the existence of the images of functions -- ...
    (sci.logic)
  • Re: My last theory.
    ... Let us see how this Replacement schema works. ... in Axiom Schema 8. ... Schema only proves the existence of the images of functions -- ...
    (sci.logic)
  • Re: Godel proved maths inconsistent not incompleteness theorem
    ... replacement and the power set axiom. ... There are different versions of the axiom schema of replacement in the ...
    (sci.logic)
  • Re: Axiom of Pairing, Scheme of Replacement from others
    ... separation follows from the axiom of replacement together with the axiom ... empty set is required. ...
    (sci.math)
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