X is an element of Y, such that 2/X is in Y. Is this allowed

[Uvohtufo]

Hey guys,

I was working on something today and I was trying to formalize what I was doing, and I tried to write out:

M = { x | ( 2 / x ) $\in$ M }

Then I thought about it and wondered whether this is allowed in set theory. Its not clear that there is anything in M at all, and thus its not clear whether any X would be in M. The criterion rests on an infinite number of other criterion. Is this allowed in any system of set theory? And if it isnt, why not?

If it is allowed, how about:

The Natural Numbers = {x | x-1 $\in$ M $\vee$ x = 0 }

 

[Fredrik]

M = { x | ( 2 / x ) $\in$ M }

This doesn't really make sense, since you haven't said anything that ensures that the expression 2/x makes sense. You could attempt to remedy that by writing e.g.
$$M=\{x\in\mathbb Z\,|\,2/x\in M\}$$ instead. But this still doesn't make sense, since you're using the definition of M in the definition of M.

{x | x-1 $\in$ M $\vee$ x = 0 }

The set of all x such that x-1 is in M and x=0... This would be {0} if -1 is in M, and ∅ otherwise.

 

[MathematicalPhysicist]

As Fredrik said, it doesn't make sense. Perhaps you wanted to say something like x=2/x, but I am just guessing...

 

[micromass]

It makes sense if you don't use it as a definition. So if $M$ is already defined, then $\{x\in \mathbb{Z}~\vert~2/x \in M\}$ is well-defined and thus the equality

$$M=\{x\in \mathbb{Z}~\vert~2/x \in M\}$$

makes sense and can be true or false.

But there is no guarantee that there is a set $M$ that satisfies the equality or that this set is unique, so you can't use the equality to define $M$.

 

[Uvohtufo]

So I was working on a project related to sound harmonics. I had 12 harmonics/tones expressed as whole number fractions, all between 1 and 2, and I also wanted to have the reciprocals (multiplied into the range of 1 and 2), in the set of tones I was working with. So I had 15/8, and I wanted 16/15 expressed in my set of tones too (2 / (15/8))

I mean, there was nothing stopping me from just adding the reciprocals to what I was doing, but I tried formalizing it for fun. I probably have no way of defining this, other than just defining the set by listing its elements right?

But this still doesn't make sense, since you're using the definition of M in the definition of M.

Oh I see. So if I said something like:

P = { x | ( 2 / x ) ∈ M }

then it would be okay?

Also, some what off-topic, I am also curious why its not allowed to define something by itself. I could see how its confusing, but I am not sure why that alone should be reason not to so.

The set of all x such that x-1 is in M and x=0... This would be {0} if -1 is in M, and ∅ otherwise.

Well that was an or, not an and. x = 0, or x - 1 is in M. So {0, 1, 2, 3, 4, 5...}.

so you can't use the equality to define $M$.

What do you mean? Like, I can't do $M$ = {x | x = 4} ?

 

[Fredrik]

Oh I see. So if I said something like:

P = { x | ( 2 / x ) ∈ M }

then it would be okay?

No, but if you first define M, you an then define P by writing e.g. $P=\{x\in\mathbb R\,|\,2/x\in M\}$. The main reason why you have to specify the set that x is an element of is that 2/x doesn't make sense for arbitrary sets x.

Also, some what off-topic, I am also curious why its not allowed to define something by itself. I could see how its confusing, but I am not sure why that alone should be reason not to so.

A definition of M must tell us what set M is. If you can prove that there's exactly one set such that $M=\{x\,|\,\text{some statement about M}\}$, then you can take this equality as the definition of M.

Well that was an or, not an and.

Ah, I read it wrong then. But the expression $\{x\,|\,x-1\in M\lor x=0\}$ still has several problems. First of all, what is M? Second, there's nothing that ensures that x-1 makes sense. You could however write $\mathbb N=\{x\in\mathbb Z\,|\, x-1\in\mathbb N\lor x=0\}$. But to take this as a definition of $\mathbb N$, you must prove that there's exactly one set $\mathbb N$ that satisfies this equality.

What do you mean? Like, I can't do $M$ = {x | x = 4} ?

This equality is fine. It defines M to be the set {4}.

 

[Bacle2]

I think you're dealing with Impredicative Definitions.

See, e.g. http://www.iep.utm.edu/predicat/

 

[Uvohtufo]

Thank you Fredrik, and Bacle2.

Definitions, and descriptions, I find to be extremely interesting. But that has little to do with math or physics so I'll avoid that conversation.