# To logically prove measure theory

Can the concepts of measure theory or probability theory be derived from logic in a complete fashion? Or, are the concepts of measure theory merely proven by arguments whose forms are logical? I'm looking to gain a complete understanding of measure theory, and I wonder if that means I have to understand how number theory first is derived from first order logic, for example. What book would give me the most complete derivation of measure theory? Thanks.

Stephen Tashi
From your remarks, I'm not sure you understand the role of logic. Roughly speaking, logic involves studying or applying reliable methods of deduction. Deduction involves assuming certain things are true (i.e. taking them as "given") and concluding other things follow a consequence. Logic itself does not deal with whether the "given" things are actually true or not.

The clearest interpretation of your question is ask what set of things we must taken as "given" in order to develop measure theory. I have never seen a text that developed measure theory based only on taking the properties of the natural numbers as the "givens".

(I dont' think that number theory is derived "from first order logic". You might try making certain assumptions about numbers and applying various types of logic to derive number theory.)

From your remarks, I'm not sure you understand the role of logic. Roughly speaking, logic involves studying or applying reliable methods of deduction. Deduction involves assuming certain things are true (i.e. taking them as "given") and concluding other things follow a consequence. Logic itself does not deal with whether the "given" things are actually true or not.

The clearest interpretation of your question is ask what set of things we must taken as "given" in order to develop measure theory. I have never seen a text that developed measure theory based only on taking the properties of the natural numbers as the "givens".

(I dont' think that number theory is derived "from first order logic". You might try making certain assumptions about numbers and applying various types of logic to derive number theory.)

Actually, I'm trying to fully understand the sum rule and product rule of probability theory, or more generally, in measure theory. Hopefully, there's more to it than just to accept it as an axiom. If I recall correctily, Whitehead and Russell were able to actually derive 1+1=2 from nothing more than first order logic. They proved this in a 3 volume set called Principia Mathematica. Do I have to know Whitehead and Russell to understand the sum and product rule?

Stephen Tashi
Do I have to know Whitehead and Russell to understand the sum and product rule?

I think Whitehead and Russell's work might be considered obsolete by modern logicians, but you'd have to ask an expert to confirm that.

You are the one who defines what it means for you (yourself) to understand something. I don't know what it would take for you to understand those rules because I don't know how you define "understanding".

Some people have a vision of mathematics as collection of knowledge that is built up from very basic assumptions in a completely logical and step by step manner. It isn't taught that way. I have never known an individual person who actually had this alleged step-by-step development of mathematics clearly laid out in his mind. Many people know pieces of this fabled step by step development in detail. Many more know facts such as "so-and-so proved that...." without knowing the details of the proofs. In spite of this limitation, individuals are able to do useful work in probability theory.

Actually, I'm trying to fully understand the sum rule and product rule of probability theory, or more generally, in measure theory. Hopefully, there's more to it than just to accept it as an axiom. If I recall correctily, Whitehead and Russell were able to actually derive 1+1=2 from nothing more than first order logic. They proved this in a 3 volume set called Principia Mathematica. Do I have to know Whitehead and Russell to understand the sum and product rule?

$1+1=2$ can be proved by giving some axiomatization of arithmetic (say Peano arithmetic, "PA") in first order logic ("FOL"). FOL is a language which says pretty much nothing unless you give it some axioms to work with. Well, to be fair, either you add some set of axioms A, or you prepend every theorem you're interested in with $A\Rightarrow\dots$. In the former case you call it an axiom, but not in the latter one (you may call it a definition if you like). But the main point is that if you do not have A somewhere then you cannot say anything. So, to sum up: FOL does not entail $1+1=2$, but FOL+PA does entail $1+1=2$, and FOL does entail $PA\Rightarrow1+1=2$.

Note that "axioms" here are actually nothing more than definitions. I.e. if you want to talk about numbers, then you should first define them. The logical status of an axiom and the logical status of a definition are actually very similar. So you have a similar situation with measure theory: you need of course to define the objects you are dealing with in order to give some results about them. I know at least two works dealing with the formalization (i.e. expressing in logic) of measure theory: the one by Tarek Mhamdi (http://http://dblp.uni-trier.de/rec/bibtex/conf/itp/MhamdiHT11), and the one by Johannes Holzl (http://http://dblp.uni-trier.de/rec/bibtex/conf/itp/HolzlH11). Note though, that both are expressed in higher order logic rather than first order logic (this is more a matter of comfort than need).

But, to really conclude this, I should point out that probably none of those works nor any knowledge in logic will help you get a better understanding of measure theory. Bacle2
Well, there are many schools of foundations of mathematics that may give you different answers on what math is about: formalist, intuitionist, etc. Some claim ( though I have trouble believing this) that mathematics is just a game of manipulation of symbols.

Still, I agree that logic is itself a method for deriving conclusions ( so that sentences considered true are transformed into others that are also true). But there is also , in meta-theory, a study of the relationship between provability and semantics/meaning. There are theorems relating the issues of truth and provability . A theory is considered complete if every true statement is provable. Model theory deals with the issue of meaning and truth in logic.

Still, like someone here said, you take a collection of axioms as a given, and develop conclusions from them. This is because the axioms are considered to be descriptive.
Besides, if you do not accept at a given starting point, you have the issue of the
infinite-regress, where you need to go backwards indefinitely to justify your axiom system.

Well, there are many schools of foundations of mathematics that may give you different answers on what math is about: formalist, intuitionist, etc. Some claim ( though I have trouble believing this) that mathematics is just a game of manipulation of symbols.

Well that's not really a matter of belief rather than a matter of opinion.

Still, I agree that logic is itself a method for deriving conclusions ( so that sentences considered true are transformed into others that are also true). But there is also , in meta-theory, a study of the relationship between provability and semantics/meaning. There are theorems relating the issues of truth and provability . A theory is considered complete if every true statement is provable. Model theory deals with the issue of meaning and truth in logic.

Indeed. Though I don't really see how all this helps answering the question here. Could you make the connection?

Still, like someone here said, you take a collection of axioms as a given, and develop conclusions from them. This is because the axioms are considered to be descriptive.

What do you mean by "descriptive"? The main reason why we need axioms is because we need to define the objects we want to talk about. No definition, no property. As simple as that.

Besides, if you do not accept at a given starting point, you have the issue of the
infinite-regress, where you need to go backwards indefinitely to justify your axiom system.

You're talking about the axioms of a logic here (or its inference rules, which have roughly the same ontological status). This would be our concern if we wanted to logically prove a logic. But here we want to "logically prove a theory", so the axioms of the logic are not supposed to be questionable. We are only concerned about the axioms of the theory.

lavinia
Gold Member
Can the concepts of measure theory or probability theory be derived from logic in a complete fashion? Or, are the concepts of measure theory merely proven by arguments whose forms are logical? I'm looking to gain a complete understanding of measure theory, and I wonder if that means I have to understand how number theory first is derived from first order logic, for example. What book would give me the most complete derivation of measure theory? Thanks.

I do not know a thing about logic but am curious what it would even mean to derive measure theory from logic. Can you explain that?

I do know that there is some deep connection between measure theory and set theory but have never studied it. For instance, the assumption that the Continuum Hypothesis is true implies that there is a non-Lebesque measurable subset of the unit square.

But... explain what you mean by deriving measure theory from logic. Sounds interesting.

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I do not know a thing about logic but am curious what it would even mean to derive measure theory from logic. Can you explain that?

It does not mean much actually. You can build measure theory on top of some more basic theories like set theory (but note that set theory remains a theory, not a logic). Hence the connections that can be found between them (like the one you mentionned with the Continuum hypothesis). The most basic theories (like set theory) are simply built on nothing (call it the empty theory if you want).

Note that those subsequent buildings are always achieved by adding (theory) axioms. This does not really match the usual meaning of "deriving". Deriving means simply finding some consequence of a fact. For instance you can derive measure theory-related results from the basic definitions of measure theory. But not from logic.

Logic is just the framework allowing to give a meaning to those notions of "axioms" and "derivation". But you cannot derive anything from logic itself.

Bacle2
Well that's not really a matter of belief rather than a matter of opinion.

Indeed. Though I don't really see how all this helps answering the question here. Could you make the connection?

What do you mean by "descriptive"? The main reason why we need axioms is because we need to define the objects we want to talk about. No definition, no property. As simple as that.

You're talking about the axioms of a logic here (or its inference rules, which have roughly the same ontological status). This would be our concern if we wanted to logically prove a logic. But here we want to "logically prove a theory", so the axioms of the logic are not supposed to be questionable. We are only concerned about the axioms of the theory.

Sorry, I don't know well how to use multi-quotes, so I'll comment from top-to-bottom.

1) Yes; I just meant that I personally have trouble believing someone would spend a whole career manipulating symbols they consider to have no meaning.

2)Sorry, I forgot why I stated this. I'll thiunk it thru and return.

3) I meant that the axioms reflect the properties we want in our system, e.g., the measure theory axioms reflect that we believe a measure should be countably-additive, etc.

4)Sorry, I was referring to teh axioms of the theory itself, and how we must just assume something at some point and build on it, or risk an infinite-regress (tho there are other alternatives.)

lavinia
Gold Member
It does not mean much actually. You can build measure theory on top of some more basic theories like set theory (but note that set theory remains a theory, not a logic). Hence the connections that can be found between them (like the one you mentionned with the Continuum hypothesis). The most basic theories (like set theory) are simply built on nothing (call it the empty theory if you want).

Note that those subsequent buildings are always achieved by adding (theory) axioms. This does not really match the usual meaning of "deriving". Deriving means simply finding some consequence of a fact. For instance you can derive measure theory-related results from the basic definitions of measure theory. But not from logic.

Logic is just the framework allowing to give a meaning to those notions of "axioms" and "derivation". But you cannot derive anything from logic itself.

So are you saying that you can not derive measure theory from logic but need some structures that assume some idea of sets?

It seems on the surface that the idea of a sigma algebra requires nothing but an idea of set.
But the idea of measure requires the real numbers as well.

D H
Staff Emeritus
$1+1=2$ can be proved by giving some axiomatization of arithmetic (say Peano arithmetic, "PA") in first order logic ("FOL").
Before you can prove 1+1=2 you have to have some notion of what those symbols 1, +, 2, and = mean. Certainly one way to define 2 is via $2\equiv 1+1$, in which case proving that 1+1=2 is rather trivial. It's a bit tougher with other definitions. It's extremely tedious if one uses the intuitionist approach employed by Whitehead and Russell. Intuitionism is a good way to get nowhere fast. It's an interesting way of seeing how things logically tie together after the fact (before it starts to unravel), but it is not a good approach to a subject about which one knows nothing.

But, to really conclude this, I should point out that probably none of those works nor any knowledge in logic will help you get a better understanding of measure theory.  indeed.

If you want to learn measure theory, a good place to start (IMO) is Rudin's Real and Complex Analysis. He spends the first 190 pages or so on measure theory before he delves into analysis proper.

So are you saying that you can not derive measure theory from logic but need some structures that assume some idea of sets?

Precisely.

It seems on the surface that the idea of a sigma algebra requires nothing but an idea of set. But the idea of measure requires the real numbers as well.

Indeed, it seems so (my knowledge in measure theory is very low, so I cannot confirm this for sure, but it seems very probable since few things require anything more than set theory - well, none to my knowledge).

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Before you can prove 1+1=2 you have to have some notion of what those symbols 1, +, 2, and = mean.
Indeed, and this is precisely the role of the axiomatization (typically the one of Peano as I mentionned in my post).
Certainly one way to define 2 is via $2\equiv 1+1$, in which case proving that 1+1=2 is rather trivial.
Well we generally introduce a symbol "succ" to denote the operation "+1". So that you define 2 as succ(1), and 1 as succ(0) (0 being a primitive symbol as well). Then you can define addition (typically by saying that x+0=x and x+succ(y)=succ(x+y)), and finally you can prove that 2=1+1 (or, formally, succ(succ(0))=succ(0)+succ(0)).

It's a bit tougher with other definitions. It's extremely tedious if one uses the intuitionist approach employed by Whitehead and Russell. Intuitionism is a good way to get nowhere fast. It's an interesting way of seeing how things logically tie together after the fact (before it starts to unravel), but it is not a good approach to a subject about which one knows nothing.
Well you need to go far in the development of arithmetic before intuitionism really makes things tedious. In any case, the definitions of the symbol can be achieved as easily in both intuitionistic and classical logic (since they are axioms). The difference only comes to proving things. In general, intuitionism is much more powerful than people usually think (see for instance the proof assistant Coq, which is intuitionistic, and allowed, notably, to discover the first complete proof of the 4 colour theorem).

Btw, I think Principia Mathematica was written prior to the invention/discovery of intuitionistic logic, so I do not think that Whitehead and Russel were using it. I cannot manage to verify this right now though.

Sorry, I was referring to the axioms of the theory itself, and how we must just assume something at some point and build on it, or risk an infinite-regress (tho there are other alternatives.)

That's my point: I do not see how you risk an "infinite-regress" if you talk about axioms of a theory. If you talk about the axioms of the logic, it does make sense (and it is a consequence of Godel 2nd incompleteness theorem), but for the axioms of the theory I do not see when we can have such a regress? Could you develop?

D H
Staff Emeritus
Well we generally introduce a symbol "succ" to denote the operation "+1". So that you define 2 as succ(1), and 1 as succ(0) (0 being a primitive symbol as well). Then you can define addition (typically by saying that x+0=x and x+succ(y)=succ(x+y)), and finally you can prove that 2=1+1 (or, formally, succ(succ(0))=succ(0)+succ(0)).
There are harder ways to prove 1+1=2 such as the the route taken in Principia Mathematica. See http://us.metamath.org/mpegif/pm54.43.html. Defining the successor of n as the smallest integer larger than n can make for a rather torturous proof that 1+1=2.

Well you need to go far in the development of arithmetic before intuitionism really makes things tedious.
I disagree. It doesn't take long at all. No proof by contradiction, no recursion. Intuitionism takes away what are arguably the two most powerful techniques in the mathematician's toolkit.

There are harder ways to prove 1+1=2 such as the the route taken in Principia Mathematica. See http://us.metamath.org/mpegif/pm54.43.html. Defining the successor of n as the smallest integer larger than n can make for a rather torturous proof that 1+1=2.

Interesting, I never saw that definition before. Indeed the proof is much more torturous that way!

I disagree. It doesn't take long at all. No proof by contradiction, no recursion. Intuitionism takes away what are arguably the two most powerful techniques in the mathematician's toolkit.

The nowadays-generally-used definition of intuitionism just excludes proof by contradiction, but not proof by recursion (see for instance Intuitionistic Logic @ Stanford Encyclopedia)! Indeed removing recursion would turn most things we usually consider true as undecidable. You could not even define a reasonable notion of number. On the other hand proof by contradiction is far less important, take as an evidence of this all the mathematical results that were developped in Coq.

Bacle2