Originally posted by uzlessIf you want to define your own set of mathmatical axioms that either avoid these problems with zero or even just avoid zero altogether then go ahead. Don't come crying to me when the whole lot fails to work.
Again, just because it's convenient to define something, it doesn't necessarily make it right.
You say it doesn't matter *why* because it's a *definition*...why do you take this definition on face value? Just because we are taught it's a definition and therefore irrefutable?
Is it possible this definition is wrong? Is there a way to test this definiti ...[text shortened]... axioms than our current number system. Kind of like QWERTY keyboards vs Dvorak. 😉
Originally posted by uzlessMy understanding here is a bit shady.
Ignoring Xanthos's trailer park comment for the moment, you are referring exactly to what I am getting at. My question is not necessarily whether or not 0!=1 is correct, but rather whether or not a mathematical system that we have devised requiring 0!=1 to be correct is a system we should be using in the first place.
You state mathematics is currently fou ...[text shortened]... DED it to fit his model of the Universe...turns out it was just his model that was wrong.
As far as I know, there are a few alternative philosophical stances within Set Theory that change a few things - they basically limit mathematics to certain things that can be 'made' in certain ways, that is it's mathematics with less objects and less proofs. I think mainly the things that drop off are some results with infinities. So they will have no effect on 0!=1. I don't think that 0!=1 is really considered much of a problem for mathematics, to be honest. Neither an example of a bigger problem or a problem in itself. So a few ideas that 'reduce' Set Theory I don't think really help you.
Going in the other direction, expanding Set Theory: Set Theory on the other hand *might* be generalised by something called Category Theory. Set Theory is basically predicated on the idea of putting something in something else (function mapping set to set.) Category Theory asks is there a more general way to describe this putting something in something else than the way it's done in Set Theory - and answers yes, as it turns out. It then describes a more generalised version of this than functions and sets offer. Some of the alternatives that feature in this generalization *do* change some major results in mathematics. In other words, what works for sets and functions doesn't work, if rather than sets and functions we use a different abstract system of 'putting something in something'.
However, Category Theory is not widely accepted as the foundation of mathematics - Set Theory is. (Category Theory is accepted in other ways, however, although it is borderline.) Also the movement to replace Set Theory with Category Theory at the foundation has, I believe, somewhat run aground to say the least. Further, from your perspective, I don't think Category Theory will change that 0!=1 at all. But nonetheless - to counter the Set Theory dogmatists Category Theory certainly provides something to think about.
As you say, such things might require some thought. Familiarize yourself with Set Theory for starters, and then good luck thinking your way beyond it 🙂
The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC.
An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, such as numbers, must be subsequently defined in terms of sets.
The ten axioms of ZFC are listed below. (Strictly speaking, the axioms of ZFC are just strings of logical symbols. What follows should therefore be viewed only as an attempt to express the intended meaning of these axioms in English. Moreover, the axiom of separation, along with the axiom of replacement, is actually a schema of axioms, one for each proposition). Each axiom has further information in its own article.
1. Axiom of extensionality: Two sets are the same if and only if they have the same elements.
2. Axiom of empty set: There is a set with no elements. We will use {} to denote this empty set.
3. Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
4. Axiom of union: Every set has a union. That is, for any set x there is a set y whose elements are precisely the elements of the elements of x.
5. Axiom of infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.
6. Axiom of separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds.
7. Axiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements.
8. Axiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.
9. Axiom of regularity (or axiom of foundation): Every non-empty set x contains some element y such that x and y are disjoint sets.
10. Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.
The axioms of choice and regularity are still controversial today among a minority of mathematicians. Other axiom systems for set theory are Von Neumann-Bernays-Gödel set theory (NBG), the Kripke-Platek set theory (KP), Kripke-Platek set theory with urelements (KPU) and Morse-Kelley set theory. All of these are axiomatic set theories closely related to ZFC: axiomatic set theories with quite different approaches are (for example) New Foundations and systems of positive set theory.
http://en.wikipedia.org/wiki/Axiomatic_set_theory
Originally posted by TommyCNow we're getting somewhere...
My understanding here is a bit shady.
As far as I know, there are a few alternative philosophical stances within Set Theory that change a few things - they basically limit mathematics to certain things that can be 'made' in certain ways, that is it's mathematics with less objects and less proofs. I think mainly the things that drop off are some results wit ...[text shortened]... self with Set Theory for starters, and then good luck thinking your way beyond it 🙂
