Russell's and Frege's Definition of Number

Hurkyl

In the definition AlephZero was referring to, "set with one element" cannot be parsed into smaller pieces -- the sequence of three characters "one" doesn't have any meaning of its own in that phrase.

The definition of "X is a set with one element" is pretty simple -- it is the conjunction of:

• There exists x such that x is in X
• For all x and y such that x is in X and y is in X, x = y

i.e. [itex]\left( \exists x: x \in X \right) \wedge \left( \forall x \in X: \forall y \in X: x = y \right)[/itex] (or some equivalent thereof)

guigus

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 class of similar classes. That is, a number is a class containing only classes that are similar to each other. Then, you have to define similarity without resorting to the concept of a number. By which the one-to-one correspondence between the elements of any two classes will not suffice, as it still depends on the numbers two and one. Meanwhile, Russell's definition remains correct. And if you repute it as wrong, then you must find a number that does not fit it.

guigus

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.

disregardthat

The definition does not use the concept of number to define a number. He first defines, for any class, the "number of a class" as the class of all classes that are similar to it (presumably bijectively), and then goes on defining a "number" as anything that is a "number of a class". Succinctly; he states that a number is an equivalence class (under bijection) of classes. There is no circularity here (nor pedagogy, but I'll leave it at that).

Sure it is rigorous insofar his system of axioms is, but it is not a definition of what a number is. It is a definition of what he calls "numbers" with respect to his axiomatic system of classes, but it does not encapsulate the general concept of a number. It was proposed, as I understood OP, as a universal definition of what it means to be a number. That simply cannot be done.

Compare with the concept of a function. It has many definitions in different axiomatic settings, but they don't contradict each other, nor are they battling for the status as the "correct" definition of a function. They simply reflect how a function can be treated in the particular setting one find oneself.

The concept of a number, and a function, remains a part of language, not logic.

Hurkyl

My apologies. When I said "cannot be parsed into smaller pieces", I was assuming we were not considering parsing things in an incorrect fashion -- such as parsing (as (possibly meaningless) English) the English phrase that consists of the same letters in that order.

guigus

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 one, since it means the number of elements of a set -- the set with one element -- and if it stops meaning that, then the sentence becomes meaningless.

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 be done": you must prove what you say, by showing a number that doesn't fit Russell's definition.

Logic itself goes far beyond "logic," and it certainly can, just as language can, define a number. And Russell's definition goes far beyond what Russell himself has imagined, as I can show you, if you wish. And yes, it defines what a number is, you just have to really understand it.

Hurkyl

But what AlephZero defined wasn't an English phrase. The English phrase you want to consider and the technical mathematical term AlephZero referred to are merely homonyms.

Of course, the name was chosen to be suggestive -- the technical term is meant to be logically equivalent to something we might mean by the English phrase.

The technical term doesn't define "set with ____ element(s)" where ___ is replaced by some sort of number. It only defines "set with one element".

English dictionaries are not mathematical reference books.



(moderator's hat on) Anyways, this discussion had been dead for 5 years, is in the wrong section, and the necromancer seems to just want to be contrary, so I don't see any reason to leave the thread open.