Re: A new definition for Cardinality

Daryl McCullough wrote:
MoeBlee says...

I'm not sure whether meeting the criteria of eliminability is
sufficient for preserving consistency; I think that perhaps also the
criteria of non-creativity must be satisfied.

I'm not sure I understand the distinction you are making. What does
non-creativity mean? And what's an example of a non-non-creative
definition?

Let the source language be the language of the source theory. Let the
extended language be the language made by adding a new symbol to the
set of symbols of the source language. Let the extended theory be the
theory made by adding the definitional axiom for the new symbol to the
source theory.

The criterion of eliminability is that, for any formula of the extended
language, there is a formula of the source language that is, in the
extended theory, equivalent to the formula of the extended language.

The criterion of non-creativity is that, for any formula of the source
language that is a theorem of the extended theory, the formula is also
a theorem of the source theory.

That's as succinct as I can put it, but for a rigorous formulation, I'd
use symbols of the meta-language, such as the turnstile symbol.

I'll try to see if I can come up with an example of a formula that
violates the criteria of non-creativity but doesn't violate the
criteria of eliminability.

By the way, another poster mentioned that this is due to Padoa. Padoa's
method is for proving that a symbol is independent, per a theory, from
the other symbols of the language of the theory. But it was Lesniewski
who came up with the criteria of eliminability and non-creativity.

MoeBlee

.

Relevant Pages

  • Design of a URL encoded language to specify sets of files on a WebDAV server
    ... I'm currently in the process of designing a language which will be used ... There will be a more verbose query language also ... One way of specifying search criteria in MetaFS is by using a dynamic ... Boolean Logic Engine ...
    (comp.lang.python)
  • Re: Factoring more beautiful now
    ... indistinguishable by every criteria. ... The Horse is Brown. ... Language does not. ...
    (sci.crypt)
  • Re: Factoring more beautiful now
    ... indistinguishable by every criteria. ... The Horse is Brown. ... language is not directly comparable to logic. ...
    (sci.crypt)
  • Re: what software should I use?
    ... >> criteria in a simple to use graphical user interface. ... > Download Ruby, and write a Tk interface in a few lines. ... > Modern languages come with all these common stuff bundled with the language. ...
    (comp.programming)
  • Re: A new definition for Cardinality
    ... violates the criteria of non-creativity but doesn't violate the ... criteria of eliminability. ... Daryl, I have not come up with an example. ... It would be interesteing if elminability implied non-creativity. ...
    (sci.math)