Re: Is the empty set a number?

On 7 Apr, 23:13, "Peter Webb" <webbfam...@xxxxxxxxxxxxxxxxxxxxxxxxx>
wrote:
<jonas.thornv...@xxxxxxxxxxx> wrote in message

news:9fd1146e-a546-4cb9-9cf6-9f5a7d744c6d@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On 7 Apr, 18:13, Randy Poe <poespam-t...@xxxxxxxxx> wrote:

On Apr 7, 12:08 pm, jonas.thornv...@xxxxxxxxxxx wrote:

Well i want you to prove that 0 is a number

OK. Let's start with your definition of "number"
and your definition of "0". What do you mean
by "0"? What do you mean by proving that
something is a number?

- Randy

I do not mean anything by 0 i have been taught that 0 is a number of
no "value"
A number is something that is necessary to represent a "value"
For me is absense of a value not a number, if x=0 4x seems to be a
nonsensical expression.

********************
In answer to the question in your subject line, the empty set is not a
number. The empty set is a set, and whilst you haven't defined "number" I
would assume that you do not consider it as a set - this seems incompatible
with your "definition" of a "number" as a "value".

Well 21 is a number and can be described like an ordered set in
standard set theory or am i wrong in assuming that(2,1) is a set
theoretic equivalent to 21?

Why do you say that an ordered set can not represent a value i do not
understand.

What logicians have done is to show that the structure which commences with
{} and creates other sets using the rule S(x) = x U {x} is isomorphic to the
definition of numbers given by Peano where S(y) = y+1.

This is over my current math skills

Then, by proving results about sets, we can prove results about numbers.

Sure i consider ordered set to have numerical value

Note that this definition of S(x) = x U {x} is arbitrary; other mappings are
possible,
including S(x) = {x}.
Note that this isomorphism also does not require {} <-> 0.
The same thing would work if we defined the isomorphism as

{} <-> 1   (with the disadvantage we have no direct way of expressing zero)

or

{{}} <-> 0  (with the disadvantage that we have to type additional brace
characters all the time)

So for reasons of simplicity, the most commonly used  isomorphism is based
upon

{} <-> 0

So no, {} is not a number. It can be associated with a number (as defined by
Peano) through a straightforward isomorphism, but equally other sets could
be mapped to zero, or {} could be mapped to some other number, and the
isomorphism would still hold. Its a logical but arbitrary choice that we
usually create the isomorphism using {} to represent 0.

Hope this helps

Well thanks, but it was a bit over my knowledge in set theory and
math.
Can you tell me if two empty sets equals one empty set i would be
greatful for an answer.

Your <-> is it a not equal sign?

Peter Webb

.

Relevant Pages

  • Re: Is the empty set a number?
    ... The empty set is a set, and whilst you haven't defined "number" I ... Then, by proving results about sets, we can prove results about numbers. ... Note that this isomorphism also does not require 0. ... Peano) through a straightforward isomorphism, ...
    (sci.math)
  • Re: Is the empty set a number?
    ... In answer to the question in your subject line, the empty set is not a number. ... Note that this isomorphism also does not require 0. ... It can be associated with a number (as defined by Peano) through a straightforward isomorphism, but equally other sets could be mapped to zero, or could be mapped to some other number, and the isomorphism would still hold. ... Its a logical but arbitrary choice that we usually create the isomorphism using to represent 0. ...
    (sci.math)
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