Re: an important set theory post

On Sun, 27 Jul 2008 07:12:46 -0700 (PDT), calvin
<crice5@xxxxxxxxxxxxxx> wrote:

On Jul 27, 9:56 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Sat, 26 Jul 2008 13:48:17 -0700 (PDT), calvin
...
That epsilon symbol is just a synonym for
"is an element of". And "is an element of" _is_
taken to be undefined in set theory.

Strange, since everyone knows what it means.

At some point it may penetrate: We're talking
about mathematics as it is. Whether or not
something seems strange to you doesn't matter.
And whether or not "everyone" _thinks_ they
know what something means doesn't matter either.

The "everyone knows what it means" notion of
"is an element of" is not precise enough to serve
as a mathematical definition.

... we already have a definition of set.

No, we don't have a definition of "set" either.

Well, it wasn't too long ago in this forum that
I referred to the 'set of natural numbers', and
someone jumped in to say that the natural numbers
are not a set. That made it look like there must
be a definition of a set somewhere.

First, anyone who says the natural numbers do not
form a set is either very confused or is using a very
non-standard set theory. The natural numbers
are certainly a set in ZF.

It does not follow that we have a definition of "set".
Instead we have axioms _about_ sets. The axioms
imply that there is a set N such that x is an element
of N if and only if x is a natural number.

And what needs to be defined is the _relation_
"this is an element of that", not the word
"element" in isolation.

An 'element', it seems to me, is something that exists.

Accepting that as a definition of "element" for the
sake of argument, that doesn't say anything about
what "is an element of" means.

But I addressed that too. You ignored what came later.

Here's a question for you: Is it possible to define the
meaning of _every_ word we use?

Yes, but, when we use words to define words, at
some point it becomes circular.

Circular definitions are no definition at all.
The official definitions are required to be
non-circular. _Hence_ there must be
_something_ which is left undefined
(because the terms used in the _first_
definition cannot have been defined in
previous definitions, since at that point
there are no previous definitions).

_Something_ _must_ be undefined in a formal
basis for mathematics. It's been found that
if we take "set" and "is an element of" as undefined
then that's enough - we can define everything
else we need in terms of those two undefined
primitives.

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

  • Re: an important set theory post
    ... Every term in mathematics has a definition. ... exception is the symbol for "is an element of" from set theory. ... Letters are elements of my mailbox. ... "Understanding Godel isn't about following his formal proof. ...
    (sci.math)
  • Re: Cantor Confusion
    ... But so far all the various objects of mathematics can be ... That is what is meant by set theory being a foundation ... functions, limits, calculus, etc. all can be founded in set theory, it would have to ... Not that there is anything wrong with strange, ...
    (sci.math)
  • Re: all the incompleteness proofs are worthless untill...
    ... godel thought the system he usesd and AR where valid- which russell ... The development of mathematics in the direction of greater exactness has ... system of Principia Mathematica 2 and, on the other, the axiom system ... for set theory of Zermelo-Fraenkel.3 ...
    (sci.logic)
  • Re: Cantor Confusion
    ... Han de Bruijn wrote: ... But so far all the various objects of mathematics can be ... That is what is meant by set theory being a foundation ... Not that there is anything wrong with strange, ...
    (sci.math)
  • Re: Cantor Confusion
    ... But so far all the various objects of mathematics can be ... That is what is meant by set theory being a foundation ... functions, limits, calculus, etc. all can be founded in set theory, it would have to ... Not that there is anything wrong with strange, ...
    (sci.math)
Loading