1

It appears to me that a fair number of issues with allowing ZFC to work with other mathematical topics is that one cannot phrase certain definitions inside ZFC. Would not this be fixed by allowing definitions to be creative? If we decided that we could define anything we want, and implicitly state that every theorem about this definition has the disclaimer "if definition exists, then...".

I'm mainly wondering what issues crop up if you take this stance on definitions.

Asaf Karagila
  • 393,674
Nethesis
  • 3,986
  • 2
    I fear, I do not understand fully. Can you elaborate more or give a very concrete example, where is the problem is, how to fix it and what exactly a creative definition is? This would be very helpful for me. – Imago Mar 10 '15 at 07:36
  • A "creative" definition is an axiom ... We cann add to the axioms of geometry the new axiom : "there exists four-sided triangles" and then we are free to use this axiom for proving theorems about four-sided triangles. As you suggest, all those new theorems will be "true" in a domain satisfying the new axiom, i.e. in every model where there exists a four-sided triangle. – Mauro ALLEGRANZA Mar 10 '15 at 07:42
  • Well in the book in reading it states that a definition is merely an abbreviation for a formula in the object language. This abbreviation must meet the criteria of non creativity, so essentially it should not "do" anything that cannot be done somehow with axioms. I feel that this is somewhat restrictive, and that we might be better placed by allowing definitions that can describe any formula, however their existence is not assumed. This should allow for greater abstraction by removing the need to show that, for instance, an abstract group exists. – Nethesis Mar 10 '15 at 07:45
  • I'm aware that that particular example may not be great – Nethesis Mar 10 '15 at 07:47
  • But we do not need to assume a definition describes an object which exists to logically follow consequences from it. We may define a "four sided" triangle and show what follows from it, so that if, in some strange distant future, we find that one exists, all of these properties of it already lie proven. – Nethesis Mar 10 '15 at 07:53
  • If every theorem from such definitions is of the form "if A then B" then there is no immediate issue I can see, for if it turns out our object cannot exist, the theorems will still be true - we will simply have "not A". – Nethesis Mar 10 '15 at 07:55
  • 1
    I agree; the "convention" is that we call them axioms and not definitions... But "formally" a definition is an axiom, i.e. we introduce a new symbol in theory expanding its language; this new symbol is introduced thorugh an axiom which is a bi-conditional : "new-simbol (the definiendum) $\leftrightarrow$ definiens" – Mauro ALLEGRANZA Mar 10 '15 at 08:20
  • @Nethesis: this is precisely why I made a post about the fact that we need a theory of theories. It is terrible that FOL doesn't allow you to define formally what is a definition or a property, and therefore forbids you to study them and how they interact. Note that in order to not be crushed by diagonal argument, you would have to restrict the quantifications over propositions and definitions: a solution is to stratify syntax. Moreover, it is perfectly imaginable that some concept "exists" or makes sense in an informal sense, but has no model (or only the empty set) in set theory. – sure Mar 10 '15 at 09:34
  • @sure - It is not so; see George Tourlakis, Lectures in Logic and Set Theory. Volume 1 : Mathematical Logic (2003), page 112-on : Ch.I.7. Defined Symbols, for the FOL treatment of definitions. – Mauro ALLEGRANZA Mar 10 '15 at 12:10
  • An you can see this post for a summary. – Mauro ALLEGRANZA Mar 10 '15 at 12:13
  • 1
    The trouble with "just define everything you want" is that some definitions simply don't work, and it's not obvious ahead of time which definitions those are. People show up here every month asking about "the largest real number less than 5", which sounds very plausible, but there is no such number. Similarly you can say "The giant nineteen-legged purple water buffalo that lives at Madison Square Garden", but there is no such thing, so proving theorems about it is a waste of time. – MJD Mar 10 '15 at 13:35
  • Definitions conform to the same rules of syntax as any other statement. So, you can tentatively play around with different "definitions" within ZFC and FOL introducing them as an assumption (or premise) at the beginning of a proof and seeing where they take you. – Dan Christensen Mar 10 '15 at 15:06
  • @MJD would it not be correct to say that definitions could in principle stand by themselves, without whatever they are definitions of needing to exist? By this I mean, that I can define an asymptotic d-module (made up) as whatever I'd like, and prove theorems about these modules, but there is nowhere where I could intersect these theorems with those about objects which are well-behaved unless I either (a) Find a well behaved object which meets the definition or (b) Assume that there exists an object which meets the definition. – Nethesis Mar 10 '15 at 15:43
  • In this way I could perhaps define a group - I admit this may not be required for this definition - without ever having to show off an example of a group. For if I ever use group theory theorems in other areas of maths, I must have found something which meets the criteria to be a group, and so the theorems would be fine. I could define a set which is paradoxical, and derive theorems about said set, but I could not ever use these theorems elsewhere without assuming (wrongly) that this set exists. Am I making any sense..? I can try to clarify what I mean if you give me time – Nethesis Mar 10 '15 at 15:45
  • If you're going to define a D-module as a finite nonabelian division ring, don't you think you would want to know if any D-modules actually exist before you try to export your theory of D-modules to other branches of mathematics? Developing a theory of D-modules could be a lot of work, and it will all be wasted if there aren't any D-modules, or if it turns out that your theory proves not only true things but also false things. – MJD Mar 10 '15 at 15:47
  • @MJD I was hoping that there was no such thing as a D-module, I'll think a bit more before making up a name. – Nethesis Mar 10 '15 at 16:19
  • Anyway, I do see what you mean, I was just wondering whether or not there was any specific reason found that I could not do what I suggested. It would be interesting to invent definitions an theorems for things which may or may not exist, in the hope that perhaps some example may appear later. I mean...we don't give an example for every single type of group, do we? Or has it been proven that every single type of group "exists" in a well defined sense? – Nethesis Mar 10 '15 at 16:21
  • If there is such a thing as a D-module, I have not heard of it. – MJD Mar 10 '15 at 17:17
  • @MJD Ah, I thought you indicated you had, In any case http://en.wikipedia.org/wiki/D-module – Nethesis Mar 10 '15 at 18:04
  • @MJD, Nethesis There is no requirement that an object exist satisfying the property you've defined in order to figure out what other properties such an object would have if it existed. A great example of this is the theory of inaccessible cardinals; you can show that the cumulative hierachy of rank an inaccessible cardinal forms a Tarski's class and is an inner model of ZFC, within ZFC - even though you have no idea throughout the proof whether any inaccessible cardinals actually exist. Almost all large cardinal theory works this way. – Mario Carneiro Mar 10 '15 at 18:44
  • @MarioCarneiro Thank you – Nethesis Mar 11 '15 at 08:33

1 Answers1

4

Of course you are allowed to do whatever you want. It might just be provable that what you want is inconsistent.

More specifically, you can say something like "If $\varphi$ defines a set, then such and such and such". It just might be that $\varphi$ does not define a set, or it might depend on additional assumptions (e.g. the class of ordinals smaller than the least inaccessible cardinal might be a set or a proper class).

And in some sense this is what set theorists do. They assume more axioms, which in turn gives us more information about what sort of sets - strange as they might be - exist, and we see what sort of consequences these axioms have and how they fit together in all kind of ways.

The problem arises when you want to talk about things like "Assume the set of all sets exists, then ...", and then the assumption is always false in $\sf ZFC$, so it doesn't even make an interesting assumption. So in order to overcome these sort of problems in category theory we have universes that allow us to slightly modify the meaning of the word "set" (e.g. sets are element of this universe, and then they are elements of a larger universe).

But even then, there's no really problem with being creative in the definitions, the question is just whether or not your creativity is worthwhile. I can imagine a world where I can walk upside down from the ceiling; but it's not this world so it's nothing more than a dream. You can dream about a world where the ordinals make a set in $\sf ZFC$ but it's not going to happen, unless $\sf ZFC$ is inconsistent.

Asaf Karagila
  • 393,674