When you say that something is a theorem, is that merely your philosophical commentary about math?

[PseuTonym]

It seems to be a standard practice in mathematics to write some statement and assert that it is a theorem or label it with the word “theorem.” So, the question in the title of this thread might be surprising if it were asked during a lesson or lecture that is officially part of a mathematics course in some institution of formal education.

Ordinary language sometimes sends a signal that we are dealing with something studied in the early grades of elementary school. For example, it would not be unusual to write:

35 is an odd number.

We have an opportunity to use slightly more formal mathematical symbolism by writing something like:

Odd(35)
Definition: Odd if and only if (n is an integer and there does not exist m such that n = 2m)

We can write “Odd(35)” or we can write “35 is an odd number”, and it doesn’t matter which of those two forms of language we choose in this case.

However, it seems that ordinary language (rather than mathematical symbolism) sometimes sends a signal that we are using metalanguage to assert something about a particular formal system. Apparently, contradictions can arise if perfectly legitimate meta-theorems are asserted below the meta level, and combined with ordinary theorems. So the signalling can be important for mathematical reasons.

If you answer “No” to the question in the title of this thread, then we might consider introducing a mathematical notation such as:

Thm(P) if and only if P is a theorem

Such a notation might be useful if it helps us to formulate some idea that is otherwise difficult or impossible to formulate in mathematical symbolism.

Of course, it is important to understand exactly what is the idea that is formulated. We need to distinguish between asserting that a mathematical claim is true, and asserting that a mathematical claim is a theorem. According to an incompleteness theorem of Godel, the mere truth of a mathematical claim is not enough to imply that it is a theorem.

Of course, a mathematical claim is a theorem relative to some system of axioms. This relative nature of a theorem will always be clear if we begin to study what is alleged to be a proof of the theorem. The proof or alleged proof will have to invoke something, such as lemmas and other theorems that are ultimately based on some axioms. There is no problem proving that an axiom is also a theorem. However, there might be a problem convincing ourselves that an alleged axiom is actually true.

 

[Ignatius]

Looks like no one answered.
A theorem is a syntactically correct expression that is deducible from the given axioms of a deductive system.

 

[Tomdstone]

If you answer “No” to the question in the title of this thread, then we might consider introducing a mathematical notation such as:

Thm(P) if and only if P is a theorem

Such a notation might be useful if it helps us to formulate some idea that is otherwise difficult or impossible to formulate in mathematical symbolism.

I don’t see why such a notation would be helpful.

 

[PseuTonym]

I don’t see why such a notation would be helpful.

I cannot guarantee that such a notation is helpful. I would like to introduce a train of thought to indicate how the notation is potentially helpful.

Consider the set of odd numbers that are greater than zero and less than one hundred. That is a very small, finite set, and there should be nothing controversial about it. Let’s give the name u to that set. If we are permitted to use three consecutive dots to indicate omission, then we can write u = {1, 3, 5, 7, 9, 11, … 99}.

Is the number 2 an element of u? In attempting to answer that question, we might pay attention to the following theorem:

Thm(2 isn’t odd)

Of course, you will not find something so trivial in Euclid’s Elements with the label “theorem.” In Euclid’s Elements, it would not even get the label “proposition.” However, we are using the word “theorem” to indicate simply that “2 isn’t odd” has been deduced from our axioms, and we don’t intend to include any connotation that the result is interesting, difficult or important.

For convenience, let us give the name P(t) to the property (t is an integer) and (t is odd) and (t is greater than 0) and (t is less than 100). When we replace the variable t with a specific value (such as 2), P(t) becomes a statement (such as P(2)).

The reason that we are interested in Thm(2 isn’t odd) is that it implies the following:
Thm(not P(2))

We might now jump to the conclusion that 2 isn’t an element of u. Let’s try to fill in the gap and see exactly what process was used to reach the conclusion that 2 isn’t an element of u.

It seems that, for any value of t, if Thm(not P(t)), then t isn’t an element of u.
We can now apply a basic rule of inference known as “modus ponens”: if we have deduced A and if we have also deduced (if A then B) then we are entitled to deduce B.

Note that, although this rule of inference is conceptually analogous to the well-formed formula (((A and (A implies B)) implies B) of propositional logic, there is a difference between a well-formed formula and a rule of inference. The rule of inference allows us to detach the final statement B from the rest of the well-formed formula, and to assert that we have deduced B.

Now, can you imagine somebody (perhaps named ZF) telling you that there is a property Q(t) and a set w such that w = {t: Q(t)} and that, for some set b, we reach the conclusion that b isn’t an element of w, without using modus ponens, and that we also reach the conclusion Thm(Q(b))? That isn’t a typographical error. You would expect that we need to make use of Thm(not Q(b)). Instead of making use of the theorem that Q(b) is false to reach the conclusion that b isn’t an element of w, we use the conclusion that b isn’t an element of w to deduce the theorem that Q(b) is true. Surely this would be a very strange scenario.

The strangeness of that scenario might motivate us to ask a question.
Remember our set u and our property P(t). Obviously 3 is an odd integer between zero and one hundred, and obviously 3 is an element of u. Surely we agree that, for any value of t, if Thm(P(t)), then t is an element of u. Surely we use that, along with Thm(P(3)) and modus ponens, to arrive at the conclusion that 3 is an element of u. Can you imagine that we might not need to use Thm(P(3))? Can you imagine that we would instead use Thm(3 is an element of u) to reach the conclusion Thm(not P(3))?

Now, ZF tries to console us, and tells us that we don’t have to worry about that happening because, in the strange scenario of w and Q(t), we reach the conclusion that b isn’t an element of w. Our question is different because our question is about reaching the conclusion that 3 is an element of u.

Of course, we have to acknowledge that it is different, but surely there is an analogy. It is going to seem almost painfully obvious to describe the analogy in detail, but let us give it a try:

Earlier, we reached the conclusion that 2 isn’t an element of u.
Now we wish to reach the conclusion that 3 is an element of u.
Earlier, we used the result Thm(not P(2)), modus ponens, and the assumption that for any value of t, if Thm(not P(t)), then t isn’t an element of u.
Now, we use the result Thm(P(3)), modus ponens, and the assumption that for any value of t, if Thm(P(t)), then t is an element of u.

ZF tells us, “Yes, you have an analogy there. However, what you have there is incorrect. If you had a correct analysis of the concepts, the analogy would disappear.”

“Furthermore,” ZF tells us, “you are keeping yourself ignorant when you try to use the concept Thm to formulate your ideas. Let Cons be Godel’s number-theoretic sentence that can be interpreted as asserting its own unprovability. Let z be the following set: {t: (t = 3 and Cons) or (t = 2 and not Cons)}.
If our system of axioms for number theory isn’t consistent, then z = {2}.
If our system of axioms for number theory is consistent, then z = {3}, and the mere truth of Cons is sufficient for us to reach the conclusion z = {3}. Godel showed that if our system of axioms for number theory is consistent, then we are never going to be able to reach the conclusion Thm(Cons). So your approach will never allow you to reach the conclusion that z = {3}.”

Now, what do you think? Is a notation such as Thm potentially helpful? I used a bit of rhetoric with an imaginary character named ZF, because I thought that some imaginary dialogue might make things more interesting. However, my question isn’t intended to be rhetorical. Please let me know what you think.

 

[Tomdstone]

I cannot guarantee that such a notation is helpful. I would like to introduce a train of thought to indicate how the notation is potentially helpful.

Consider the set of odd numbers that are greater than zero and less than one hundred. That is a very small, finite set, and there should be nothing controversial about it. Let’s give the name u to that set. If we are permitted to use three consecutive dots to indicate omission, then we can write u = {1, 3, 5, 7, 9, 11, … 99}.

Is the number 2 an element of u? In attempting to answer that question, we might pay attention to the following theorem:

Thm(2 isn’t odd)

Of course, you will not find something so trivial in Euclid’s Elements with the label “theorem.” In Euclid’s Elements, it would not even get the label “proposition.” However, we are using the word “theorem” to indicate simply that “2 isn’t odd” has been deduced from our axioms, and we don’t intend to include any connotation that the result is interesting, difficult or important.

For convenience, let us give the name P(t) to the property (t is an integer) and (t is odd) and (t is greater than 0) and (t is less than 100). When we replace the variable t with a specific value (such as 2), P(t) becomes a statement (such as P(2)).

The reason that we are interested in Thm(2 isn’t odd) is that it implies the following:
Thm(not P(2))

We might now jump to the conclusion that 2 isn’t an element of u. Let’s try to fill in the gap and see exactly what process was used to reach the conclusion that 2 isn’t an element of u.

It seems that, for any value of t, if Thm(not P(t)), then t isn’t an element of u.
We can now apply a basic rule of inference known as “modus ponens”: if we have deduced A and if we have also deduced (if A then B) then we are entitled to deduce B.

Note that, although this rule of inference is conceptually analogous to the well-formed formula (((A and (A implies B)) implies B) of propositional logic, there is a difference between a well-formed formula and a rule of inference. The rule of inference allows us to detach the final statement B from the rest of the well-formed formula, and to assert that we have deduced B.

Now, can you imagine somebody (perhaps named ZF) telling you that there is a property Q(t) and a set w such that w = {t: Q(t)} and that, for some set b, we reach the conclusion that b isn’t an element of w, without using modus ponens, and that we also reach the conclusion Thm(Q(b))? That isn’t a typographical error. You would expect that we need to make use of Thm(not Q(b)). Instead of making use of the theorem that Q(b) is false to reach the conclusion that b isn’t an element of w, we use the conclusion that b isn’t an element of w to deduce the theorem that Q(b) is true. Surely this would be a very strange scenario.

The strangeness of that scenario might motivate us to ask a question.
Remember our set u and our property P(t). Obviously 3 is an odd integer between zero and one hundred, and obviously 3 is an element of u. Surely we agree that, for any value of t, if Thm(P(t)), then t is an element of u. Surely we use that, along with Thm(P(3)) and modus ponens, to arrive at the conclusion that 3 is an element of u. Can you imagine that we might not need to use Thm(P(3))? Can you imagine that we would instead use Thm(3 is an element of u) to reach the conclusion Thm(not P(3))?

Now, ZF tries to console us, and tells us that we don’t have to worry about that happening because, in the strange scenario of w and Q(t), we reach the conclusion that b isn’t an element of w. Our question is different because our question is about reaching the conclusion that 3 is an element of u.

Of course, we have to acknowledge that it is different, but surely there is an analogy. It is going to seem almost painfully obvious to describe the analogy in detail, but let us give it a try:

Earlier, we reached the conclusion that 2 isn’t an element of u.
Now we wish to reach the conclusion that 3 is an element of u.
Earlier, we used the result Thm(not P(2)), modus ponens, and the assumption that for any value of t, if Thm(not P(t)), then t isn’t an element of u.
Now, we use the result Thm(P(3)), modus ponens, and the assumption that for any value of t, if Thm(P(t)), then t is an element of u.

ZF tells us, “Yes, you have an analogy there. However, what you have there is incorrect. If you had a correct analysis of the concepts, the analogy would disappear.”

“Furthermore,” ZF tells us, “you are keeping yourself ignorant when you try to use the concept Thm to formulate your ideas. Let Cons be Godel’s number-theoretic sentence that can be interpreted as asserting its own unprovability. Let z be the following set: {t: (t = 3 and Cons) or (t = 2 and not Cons)}.
If our system of axioms for number theory isn’t consistent, then z = {2}.
If our system of axioms for number theory is consistent, then z = {3}, and the mere truth of Cons is sufficient for us to reach the conclusion z = {3}. Godel showed that if our system of axioms for number theory is consistent, then we are never going to be able to reach the conclusion Thm(Cons). So your approach will never allow you to reach the conclusion that z = {3}.”

Now, what do you think? Is a notation such as Thm potentially helpful? I used a bit of rhetoric with an imaginary character named ZF, because I thought that some imaginary dialogue might make things more interesting. However, my question isn’t intended to be rhetorical. Please let me know what you think.

No, because if Cons is true then z=3 because you didn’t say that Cons would have to be a theorem in your definition of z.

 

[PseuTonym]

No, because if Cons is true then z=3 because you didn’t say that Cons would have to be a theorem in your definition of z.

My definition of z relied upon the notation z = {t: R(t)}. There is nothing to be gained by debating the meaning of a word or notation. So, if you insist that the notation z = {t: R(t)} has one particular fixed definition, and that it is the same definition in all possible systems of set theory, then I am not going to dispute that.

You began your reply with the word “No.” However, I did provide a train of thought for you. In fact, you quoted the whole train of thought. Do you really think that the notation Thm(P) provided no help in formulating that train of thought?

 

[Tomdstone]

My definition of z relied upon the notation z = {t: R(t)}. There is nothing to be gained by debating the meaning of a word or notation. So, if you insist that the notation z = {t: R(t)} has one particular fixed definition, and that it is the same definition in all possible systems of set theory, then I am not going to dispute that.

You began your reply with the word “No.” However, I did provide a train of thought for you. In fact, you quoted the whole train of thought. Do you really think that the notation Thm(P) provided no help in formulating that train of thought?

No. You wrote: Let z be the following set: {t: (t = 3 and Cons) or (t = 2 and not Cons)}.
You then said “Godel showed that if our system of axioms for number theory is consistent, then we are never going to be able to reach the conclusion Thm(Cons). So your approach will never allow you to reach the conclusion that z = {3}.”
I contend that this is not true because if you know that Cons is true, you will be able to say that z=3. You don’t have to know Thm(Cons).