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:
: Two sets and are said to have the same number of elements if each element of can be paired with precisely one element of in such a way that each element of is paired with precisely one element of .
Note that a prior definition of number was not given in this text. Now...
Then, in the next section, the definition of cardinal number:
: Let be any set. Define to be the collection of all sets that have the same number of elements as . We call the cardinal number of .
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 ( )is the set of all sets that have the same number of elements as ( ). HELP? I need some to explain very simply that this reasoningis not circular. THanks.