circular reasoning of the definition of "number?"

Hi. I'm going back to basics and I have been perusing the book "basic concepts of mathematics and logic" by michael gemignani. The definition 6.1 of this texts defines the phrase "same number as" as follows:

$\displaystyle Def^n\text{ }6.1$: Two sets $\displaystyle S$ and $\displaystyle T$ are said to have the same number of elements if each element of $\displaystyle S$ can be paired with precisely one element of$\displaystyle T$ in such a way that each element of $\displaystyle T$ is paired with precisely one element of$\displaystyle S$.

Note that a prior definition of number was not given in this text. Now...

Then, in the next section, the definition of cardinal number:

$\displaystyle Def^n\text{ } 6.3$: Let$\displaystyle S $be any set. Define$\displaystyle \overline{S}$ to be the collection of all sets that have the same number of elements as $\displaystyle S$. We call $\displaystyle \overline{S}$ the cardinal number of$\displaystyle S$.

Now, the text goes on to say "using the [above] definitions we could go on to develop a rigorous theory of #s as we usually think of them. I find this quite circular by way of 6.3. It states that a number ($\displaystyle \overline{S}$)is the set of all sets that have the same number of elements as ($\displaystyle S$). HELP? I need some to explain very simply that this reasoningis not circular. THanks.

Re: circular reasoning of the definition of "number?"

Why do you think it is circular? If I have the set {a, b}, surely, I can determine which sets have a one-to-one correspondence with that set. Nothing circular in that. $\displaystyle \overline{\{a,b\}}$ is the collection of all such sets (which would, of course, include {a,b} itself). I see nothing circular in that. Now, notice I did not use the term "same number of elements" which, in your quote, is purely descriptive and not a part of the mathematical definition. Perhaps that was what was bothering you - that we seem to need to determine what sets have the "same number of elements" in order to determine a "cardinal number". We **don't** we merely need to determine one-to-one correspondences which does NOT have to depend on "same number". Indeed, we can wait until **after** we have defined "cardinal number" before we even mention "same number".

Re: circular reasoning of the definition of "number?"

The concept "have the same number of elements" has the word "number" in its name, but its definition does not use numbers. If you wish, you can replace "have the same number of elements" by "have the same cardinality," "are equinumerous," "are equipollent," or "are equipotent" (these terms are taken from this Wikipedia article).

Quote:

Originally Posted by

**VonNemo19** Now, the text goes on to say "using the [above] definitions we could go on to develop a rigorous theory of #s as we usually think of them.

I don't know enough about this, but I am wondering how hard it would be to go this way. In particular, is this what Russel and Whitehead did in *Principia Mathematica*? After all, it took them 362 pages to prove 1 + 1 = 2...

Re: circular reasoning of the definition of "number?"

Quote:

Originally Posted by

**HallsofIvy** Why do you think it is circular? If I have the set {a, b}, surely, I can determine which sets have a one-to-one correspondence with that set. Nothing circular in that. $\displaystyle \overline{\{a,b\}}$ is the collection of all such sets (which would, of course, include {a,b} itself). I see nothing circular in that. Now, notice I did not use the term "same number of elements" which, in your quote, is purely descriptive and not a part of the mathematical definition. Perhaps that was what was bothering you - that we seem to need to determine what sets have the "same number of elements" in order to determine a "cardinal number". We **don't** we merely need to determine one-to-one correspondences which does NOT have to depend on "same number". Indeed, we can wait until **after** we have defined "cardinal number" before we even mention "same number".

Thank you. Very helpful.

Re: circular reasoning of the definition of "number?"

Quote:

Originally Posted by

**emakarov** ...(these terms are taken from

this Wikipedia article[/URL]).

Good article. Thank you.

Re: circular reasoning of the definition of "number?"

Quote:

Originally Posted by

**HallsofIvy** Why do you think it is circular? If I have the set {a, b}, surely, I can determine which sets have a one-to-one correspondence with that set.

With pairing? I've noticed that "pairing" was left undefined. Is there a way to define this? I know how to interpret "pairing" in terms of a function, but is there another way?