What qualities, beyond consistency, are desirable in a system of mathematics?

[PseuTonym]

For example, is omega-consistency desirable in a system of formal number theory?

First, I will try to explain what that means. Please ask questions if my explanation is not clear enough for you.

Typically, a system of formal number theory has a symbol 0 for zero, and a one-place operation symbol S where (Sn) represents what we ordinarily write as (n plus one).

Now, suppose that when we plug in a constant value for n the expression P becomes a particular sentence that is true or false for that value of n. Also, suppose that each of the following is a theorem in our system:

1. P(0)
2. P(S0)
3. P(SS0)
   etc.

(Note that the “etc” is essential, and that it stands for infinitely many theorems).

Then to say that our system is “omega-consistent” is to say that the following isn’t a theorem of our system:

“There exists n such that not P.”

That concludes the explanation.

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My motivation for starting this thread is a couple of statements from another thread:

“We already have a good set of axioms.”
Link:
forums.catholic-questions.org/showpost.php?p=13025241&postcount=22

“Any set of axioms can be used with any grammar rules - provided that there are internally consistent rules.”
Link:
forums.catholic-questions.org/showpost.php?p=13014924&postcount=3

Reading over those statements, it is interesting to consider the word “good” in the statement “We already have a good set of axioms.” Suppose that a system of formal number theory that is consistent but not omega-consistent is a good system. It would seem that a system of formal number theory that is omega-consistent is a better system.

 

[Tomdstone]

Suppose that a system of formal number theory that is consistent but not omega-consistent is a good system. It would seem that a system of formal number theory that is omega-consistent is a better system.

A better system would be a workable system. Can you give us a simple example of a workable, easily understood system of number theory which is consistent but not omega-consistent.

 

[PseuTonym]

Can you give us a simple example of a workable, easily understood system of number theory which is consistent but not omega-consistent.

That would be a good question for somebody who advocates the “consistency is good enough” point of view. I do not advocate it.

There may be a temptation when talking about theory to focus on theories in theory rather than theories in practice. Of course, theories in theory means speculation or guesses about theories. If we want to rely upon evidence rather than guesses, then we should look at facts and history.

Consider the history of logarithms. Were they invented because somebody thought that it might be possible to prove some interesting theorems using logarithms? Solving differential equations is practical in engineering. Were they invented to solve differential equations? No, the motivation was much simpler. People wanted to multiply and divide. Using tables of logarithms and anti-logarithms, it is possible to replace multiplication with the easier operation of addition, and to replace division with the easier operation of subtraction.

 

[Gorgias]

What qualities, beyond consistency, are desirable in a system of mathematics?

A good beat, and easy to dance to.

 

[Tomdstone]

That would be a good question for somebody who advocates the “consistency is good enough” point of view. I do not advocate it.

There may be a temptation when talking about theory to focus on theories in theory rather than theories in practice. Of course, theories in theory means speculation or guesses about theories. If we want to rely upon evidence rather than guesses, then we should look at facts and history.

Consider the history of logarithms. Were they invented because somebody thought that it might be possible to prove some interesting theorems using logarithms? Solving differential equations is practical in engineering. Were they invented to solve differential equations? No, the motivation was much simpler. People wanted to multiply and divide. Using tables of logarithms and anti-logarithms, it is possible to replace multiplication with the easier operation of addition, and to replace division with the easier operation of subtraction.

I don’t see the example of a system of number theory which is consistent but not omega consistent and its usefulness outside of Goedel’s ideas.

 

[PseuTonym]

I don’t see the example of a system of number theory which is consistent but not omega consistent and its usefulness outside of Goedel’s ideas.

Infinitesimals were useful before the epsilon-delta definition of limits was invented. More recently, infinitesimals have been given a formal treatment via non-standard reals. Analogously, perhaps non-standard arithmetic could have some use in future.

The following seems to be related to this topic:

“Weak theories of nonstandard arithmetic and analysis”
in Stephen Simpson, editor, Reverse Mathematics 2001, A K Peters, 19-46, 2005.

A direct link to the pdf is here:
andrew.cmu.edu/user/avigad/Papers/nonstandard2.pdf

Here is a link to the full list of documents (click on “Research”):
andrew.cmu.edu/user/avigad/

Of course, even if there is a use for non-standard arithmetic, we might nevertheless consider the most important systems of number theory to be those that are omega-consistent. So something more than mere consistency might be very important.

 

[Tomdstone]

Typically, a system of formal number theory has a symbol 0 for zero, and a one-place operation symbol S where (Sn) represents what we ordinarily write as (n plus one).

Now, suppose that when we plug in a constant value for n the expression P becomes a particular sentence that is true or false for that value of n. Also, suppose that each of the following is a theorem in our system:

1. P(0)
2. P(S0)
3. P(SS0)
   etc.

(Note that the “etc” is essential, and that it stands for infinitely many theorems).

Then to say that our system is “omega-consistent” is to say that the following isn’t a theorem of our system:

“There exists n such that not P.”

That concludes the explanation…

Let me ask the question another way. Give us an example of omega inconsistency; i.e., a case where

1. P(0)
2. P(S0)
3. P(SS0)
   etc. are all true, but it is also true that
   “There exists n such that not P.”

 

[PseuTonym]

Let me ask the question another way. Give us an example of omega inconsistency; i.e., a case where

1. P(0)
2. P(S0)
3. P(SS0)
   etc. are all true, but it is also true that
   “There exists n such that not P.”

You have replaced “theorem” with “true”, so I think that the way you have formulated your question has set you up to be disappointed with a direct answer.

If you are interested in why there exists a system that is omega-inconsistent, then the brief answer is: the compactness theorem for first-order logic tells us that such a system exists. To be precise, we should refer to it as the “compactness meta-theorem”, but if you adopt the habit of speaking the way people write, then you will likely fall into the habit of omitting the “meta.”

Here is a memory aid that you might find helpful to remember the conclusion of the compactness theorem: if a set of sentences is inconsistent, then there is a deduction of a contradiction from the set of sentences. Any such deduction is finite. Thus, any such deduction relies upon only finitely many of the sentences.

An actual proof of the compactness meta-theorem is much more involved. I looked around and found the following:
princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf

If you skim through it quickly, then you will quickly get to an example at the end under the heading “Consequences of compactness.”

 

[Tomdstone]

You have replaced “theorem” with “true”, so I think that the way you have formulated your question has set you up to be disappointed with a direct answer.

If you are interested in why there exists a system that is omega-inconsistent, then the brief answer is: the compactness theorem for first-order logic tells us that such a system exists. To be precise, we should refer to it as the “compactness meta-theorem”, but if you adopt the habit of speaking the way people write, then you will likely fall into the habit of omitting the “meta.”

Here is a memory aid that you might find helpful to remember the conclusion of the compactness theorem: if a set of sentences is inconsistent, then there is a deduction of a contradiction from the set of sentences. Any such deduction is finite. Thus, any such deduction relies upon only finitely many of the sentences.

An actual proof of the compactness meta-theorem is much more involved. I looked around and found the following:
princeton.edu/~hhalvors/teaching/phi312_s2013/compactness.pdf

If you skim through it quickly, then you will quickly get to an example at the end under the heading “Consequences of compactness.”

It looks to me like the first example is saying that there is a consistent way to adjoin the infinity symbol to the natural numbers.

 

[PseuTonym]

It looks to me like the first example is saying that there is a consistent way to adjoin the infinity symbol to the natural numbers.

In describing the model, the author is taking the liberty of using what is basically ordinary language rather than mathematical language. The point of the exercise is that there is actually no way to do that if we are restricted to creating axioms, even if we have the option of creating schemas of infinitely many axioms.

For example, suppose that we want to say that x, y, and z are any given natural numbers, and now we want to say how they are related to each other. We want to formulate some statement that we believe is true and that will be a new axiom for number theory. What can we say that will ensure that the values of the variables are restricted to the natural numbers?

Could you please comment on the following?

You have replaced “theorem” with “true”

Reading your posts, it is difficult for me to distinguish between

1. What is potentially a mistake that you are might be willing to acknowledge is a mistake,

and

1. What you have a commitment to because it is a product of your beliefs, so that it might look like a mistake to me, but you do not consider it to be a mistake.

For example:

“if an axiom is not contradictory or if it does not contradict another axiom, it would not be false. It is just an assumption.”

Link:
forums.catholic-questions.org/showpost.php?p=12818129&postcount=15

For example, in response to …

Then to say that our system is “omega-consistent” is to say that the following isn’t a theorem of our system:
“There exists n such that not P.”

… you wrote the following:

Let me ask the question another way. Give us an example of omega inconsistency; i.e., a case where

1. P(0)
2. P(S0)
3. P(SS0)
   etc. are all true, but it is also true that
   “There exists n such that not P.”

Link:
forums.catholic-questions.org/showpost.php?p=13051214&postcount=7

Do you believe that there is no distinction between the the concepts “true” and “theorem”? One problem that you might run into is the following: to do ordinary reasoning, you need to be able to get from “not true” to false. However, if we omit the parallel postulate from Euclidean geometry and we can confirm that the parallel postulate is not a theorem, and if we can get from that to the conclusion that the parallel postulate is not true, then Euclidean geometry must be inconsistent.

 

[Tomdstone]

In describing the model, the author is taking the liberty of using what is basically ordinary language rather than mathematical language. The point of the exercise is that there is actually no way to do that if we are restricted to creating axioms, even if we have the option of creating schemas of infinitely many axioms.

For example, suppose that we want to say that x, y, and z are any given natural numbers, and now we want to say how they are related to each other. We want to formulate some statement that we believe is true and that will be a new axiom for number theory. What can we say that will ensure that the values of the variables are restricted to the natural numbers?

Could you please comment on the following?

Reading your posts, it is difficult for me to distinguish between

1. What is potentially a mistake that you are might be willing to acknowledge is a mistake,

and

1. What you have a commitment to because it is a product of your beliefs, so that it might look like a mistake to me, but you do not consider it to be a mistake.

For example:

“if an axiom is not contradictory or if it does not contradict another axiom, it would not be false. It is just an assumption.”

Link:
forums.catholic-questions.org/showpost.php?p=12818129&postcount=15

For example, in response to …

… you wrote the following:

Link:
forums.catholic-questions.org/showpost.php?p=13051214&postcount=7

Do you believe that there is no distinction between the the concepts “true” and “theorem”? One problem that you might run into is the following: to do ordinary reasoning, you need to be able to get from “not true” to false. However, if we omit the parallel postulate from Euclidean geometry and we can confirm that the parallel postulate is not a theorem, and if we can get from that to the conclusion that the parallel postulate is not true, then Euclidean geometry must be inconsistent.

Right. By true in this context, I meant logically and consistently deduced from the given axioms or assumptions.

 

[PseuTonym]

Right. By true in this context, I meant logically and consistently deduced from the given axioms or assumptions.

The caveat “and consistently” in the phrase “logically and consistently deduced” is a major warning sign to me. Please explain how consistency enters into things.

If it is eventually discovered that a contradiction can be deduced from the list of all axioms and assumptions invoked and relied upon, then would you say that nothing that was deduced from that list was “logically and consistently deduced”? Alternatively, if people find a way to modify the list of assumptions so that it seems to be consistent, and so that some parts of a hierarchy of lemmas, theorems, etc can be salvaged, then will we discover that some conclusions can be “logically and consistently deduced” and are – after the discovery of a contradiction, a modification of the list of assumptions and a salvaging of some of the hierarchy of conclusions – to be recognized as things that can be “logically and consistently deduced”?

I see a serious problem with the sequence of events. It looks as though a conclusion is declared to be “logically and consistently deduced” in what is potentially some future historical era after the conclusion was merely “logically deduced.”

 

[Tomdstone]

The caveat “and consistently” in the phrase “logically and consistently deduced” is a major warning sign to me. Please explain how consistency enters into things.

If it is eventually discovered that a contradiction can be deduced from the list of all axioms and assumptions invoked and relied upon, then would you say that nothing that was deduced from that list was “logically and consistently deduced”? Alternatively, if people find a way to modify the list of assumptions so that it seems to be consistent, and so that some parts of a hierarchy of lemmas, theorems, etc can be salvaged, then will we discover that some conclusions can be “logically and consistently deduced” and are – after the discovery of a contradiction, a modification of the list of assumptions and a salvaging of some of the hierarchy of conclusions – to be recognized as things that can be “logically and consistently deduced”?

I see a serious problem with the sequence of events. It looks as though a conclusion is declared to be “logically and consistently deduced” in what is potentially some future historical era after the conclusion was merely “logically deduced.”

A consistent theory is one that does not contain a contradiction.

 

[PseuTonym]

A consistent theory is one that does not contain a contradiction.

As you know, if a system of axioms for number theory is consistent, then we have no guarantee that it is omega-consistent. If we review this discussion, then we see that the topic of consistency arises because of what you wrote earlier …

Right. By true in this context, I meant logically and consistently deduced from the given axioms or assumptions.

Thus, your concept of truth is relative to a system that you have not specified, and it could be a system of number theory that happens to not be omega-consistent.

If a system of number theory is not omega-consistent, then your concern is purely pragmatic: that the system might not be “workable” and that people might experience difficulty understanding it.

As a matter of principle, you refuse to accept omega-consistency as a restriction to be imposed on a system of number theory.

Is the above correct, or have I misrepresented your point of view?

 

[Tomdstone]

As you know, if a system of axioms for number theory is consistent, then we have no guarantee that it is omega-consistent. If we review this discussion, then we see that the topic of consistency arises because of what you wrote earlier …

Thus, your concept of truth is relative to a system that you have not specified, and it could be a system of number theory that happens to not be omega-consistent.

If a system of number theory is not omega-consistent, then your concern is purely pragmatic: that the system might not be “workable” and that people might experience difficulty understanding it.

As a matter of principle, you refuse to accept omega-consistency as a restriction to be imposed on a system of number theory.

Is the above correct, or have I misrepresented your point of view?

I think you can have an arithmetically sound, ω-inconsistent theory.BTW, I don’t see a consistent definition of omega-consistency, since Goedel has one definition, and Tarski has another.

 

[PseuTonym]

BTW, I don’t see a consistent definition of omega-consistency, since Goedel has one definition, and Tarski has another.

In this context, are you using the word “consistent” to have the same meaning as the word “standardized”? I believe that Godel introduced the terminology, so if Tarski adopted the same name to refer to at least one other concept that is slightly different, then I would not assume that there is only one concept corresponding to that name throughout the published works of Tarski. In other words, I would say something like “Tarski has others, or at least one other definition.”

This reminds me of a discussion that I had in another thread.
See this post in particular:
forums.catholic-questions.org/showpost.php?p=12701440&postcount=17
The other thread has the title “A parable: how the magnetic compass solved a linguistic, semantic, and semiotic crisis.”

The best way to guarantee that two different definitions expressed in the same language are equivalent is to simply ensure that the definitions are identical as sequences of symbols, and that the definitions of the symbols that are used in the two definitions also have identical definitions, and so on through the hierarchy of definitions down to the primitive concepts.

However, it is obviously not essential that they be expressed as identical sequences of symbols. For example, 2 times pi is equal to pi times 2, so those two different expressions could be used in two different but equivalent definitions.

Perhaps you could make a contribution to the main thread where I gathered together various ideas related to the above:

Exact Meaning versus Using Facts to Hint at Meaning
forums.catholic-questions.org/showthread.php?t=942342