Recent content by guigus

There is circularity if you use a number to define a number, as in "a number is the number of a class," although there is no circularity in "a number is an equivalence class." The same definition can be stated circularly or not, despite staying correct. It is "just saying" that "simply cannot...

It is the word "parsing" that is incorrect: we are just taking the words of an English expression separately, and they do make sense as any separate words do, otherwise dictionaries would be impossible (and the expression meaningless). The word "one" in the referred expression means the number...

You must be joking... of course "one" means the number one in that phrase. All you have to do to see that is replace those three letters by the three letters "two," and you will see how your "unparseable" sentence -- which I have just parsed -- changes.

Russell's definition, in the form you cited, is just a pedagogical one. You cannot use the concept of a number to define a number, for obvious reasons. Despite that, the definition is rigorously correct. One just needs to reformulate it in a number-free way. I already gave a first formulation: a...

If you find Russell's paradox intriguing, then wait until you really understand the definition of a number. Russell's paradox will seem a breeze.

Are you saying that the "one" in "a set with one element" is not the number one? What is it, then? If you don't know what a number is, then you don't know if it is "man-made." First you must find out what a number is, then you will be able to tell who made it, if someone. Otherwise, you will...

Russell's definition is not circular, despite appearances, but only when you succeed in formulating it without resorting to the concept of a number, which is a class of similar classes. Now you need a definition of similar classes, which are any two classes with a one-to-one correspondence...

You mean a pragmatic mathematician -- which is not necessarily a good one. A philosopher, on the other hand, will be always interested in what numbers are. This is the tautological interpretation of Russell's definition, which prevents us from truly understanding it -- a true definition does...