I don't really understand the purpose of an axiom if some laws cannot be derived from them. For example, how is one supposed to prove De Morgan's laws with only the axioms of probability?
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2In its current form, I don't understand the question. What do you mean by "laws"? What do de Morgan's laws have to do with probability? De Morgan's laws are to do with how sets work, and, besides anything else, the axioms of probability are couched in the language of set theory anyway, so that their use would assume some axiomatic system for set theory, which could presumably then be used to prove De Morgan's laws. My point is that the purpose of any particular axiom is not to derive all "laws" (theorems?), but the purpose of all the axioms together is to prove all the "laws" we want. – Will R Jan 15 '17 at 03:22
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If we are talking about propositional logic, you can simply check extensivile for all the cases. For example, we want to prove that $\overline{A\land B} = \overline{A}\lor\overline{B}$. Let $\bot$ be false and $\top$ be true you have: $\overline{\bot\land\bot}=\overline{\bot}=\top=\top\lor\top=\overline{\bot}\lor\overline{\bot}$. Then $\overline{\bot\land\top}=\overline{\bot}=\top=\top\lor\bot=\overline{\bot}\lor\overline{\top}$ (plus the symmetric case), finally $\overline{\top\land\top}=\overline{\top}=\bot=\bot\lor\bot=\overline{\top}\lor\overline{\top}$. – Bakuriu Jan 15 '17 at 09:02
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In fact, you will hardly find any contemporary mathematical work providing an explicit, self-contained axiomatic system; it's just too good to be possible in pratice. One can only hope that they state clearly all those postulates that are too technical to be considered alredy known. In Probability Theory they may take (naive) Set Theory for granted, and thus consider De Morgan's laws as something easy to prove from what you ‘already know’.
There is a special branch of mathematics, called Mathematical Logic, which is actually dedicated to investigating how modern mathematical theories can be reduced to fully-rigorous axiomatic systems (they are trying ‘formalize’ mathematics). To take a glimps of how complex this task is, have a look at this site:
http://us.metamath.org/mpegif/mmset.html
(Of course, what they do in Metamath is an extreme instance of what professional mathematical logicians are concerened with.)
Pythagoricus
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2Short answer: All laws should be derived from axioms, but it is usually impractical to do so. (Kurt Gödel is famously believed to have literally proven that there are some areas of maths that are not reducaple to even a countably infinite set of axioms, but that's a long story...) – Pythagoricus Jan 15 '17 at 00:23
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2@CortAmmon: It's a false story. "countably infinite" $\ne$ "computably enumerable". – user21820 Jan 15 '17 at 02:47
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@user21820 True, slightly inexact wording... but that doesn't mean Godel's story isn't a good once! – Cort Ammon Jan 15 '17 at 03:10
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@CortAmmon: Yea Godel's story is a good one, though in fact only a finitely long story! =P – user21820 Jan 15 '17 at 03:16
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1@user21820 Bah! We both know that the telling of his story properly is a supertask! – Cort Ammon Jan 15 '17 at 03:19
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@CortAmmon: If you forgive leaving out the β-function, there is a nice clean computability proof of the incompleteness theorems that someone came up with that I explained at http://math.stackexchange.com/a/1895288. – user21820 Jan 15 '17 at 03:21
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@user21820 First: (correct me if I'm wrong) ‘computably enumerable’ $\Rightarrow$ ‘countable’. Secondly: it's actually not a story at all! It is a highly technical result of modern Logic with serious (and to some extend controversial) philosophical implications. In order to understand what exactly are Gödel's Incompleteness Theorems and analyse how they affected the academic community in those times, you need at least a PhD. Or... you may call Dr Papadimitriou and ask him to tell a better story than me... you know, like the one in Logicomix! – Pythagoricus Jan 15 '17 at 12:34
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@Pythagoricus: You're making basic logic errors. And if you can even understand my linked post you'll see that I certainly know the details of the incompleteness theorems and their proofs. All it shows is that some theories are not computably enumerably, which does NOT imply that they are not countable!!! – user21820 Jan 15 '17 at 12:38
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Off-topic, if you wish to learn logic, you can take a look at the links in my profile. You do not need a PhD to understand the incompleteness theorems, but you will surely need a few months of consistent hard work. – user21820 Jan 15 '17 at 12:39
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@user21820 In no case am I going to argue about topics - I must admit - I have little knowledge of. Indeed you don't need a PhD to understand Gödel's proof, my point was that it is difficult to say how that discovery affected the way academics viewed mathematics, and I mentioned Logicomix to hint that there are efforts to popularise that discussion. Anyway, thank you for your comments and those links, I certainly want to know more about Logic! – Pythagoricus Jan 15 '17 at 13:03
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@Pythagoricus: Okay sorry if I came across as kind of blunt. I was slightly offended when your comment seemed to suggest that one needs a PhD to understand the incompleteness theorems. If you have any questions about logic, especially informal or philosophical ones, feel free to come to http://chat.stackexchange.com/rooms/44058/logic to discuss. I wish you all the best in your study of logic! =) – user21820 Jan 15 '17 at 13:24
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By the way, I'm not sure which Logicomix you're referring to, but there is a serious error at https://precariousimagination.wordpress.com/tag/logicomix where they claim falsely that Goldbach's conjecture is an example of an undecidable statement. – user21820 Jan 15 '17 at 14:33
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@user21820 I'm actually referring to this: https://en.wikipedia.org/wiki/Logicomix The link you provited also gives an impression of how complex this topic is... – Pythagoricus Jan 15 '17 at 14:46
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You probably can, and it's extremely rare that you have to. De Morgan's laws probably wouldn't be the worst (and they don't really involve probability), but one of the points of theorems is that we can use the ones we already have proven to prove new ones, without going all the way back to axioms. Because if we use theorem $A$ in the proof of theorem $B$, we know that given a proof of $A$ using only axioms, we could substitute that into our proof of $B$ and have a proof of $B$ using only axioms.
(I skipped over some details in an attempt to make it clearer).
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3This might answer one interpretation of the question in the title but given the body I am quite doubtful this is what is asked for. – quid Jan 15 '17 at 00:17
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In theory, every theorem in some practical formal system is by definition a sentence that can be deduced in finitely many steps by strictly following the allowed inference rules. Inference rules are more general than axioms, and commonly are of the form "If you can deduce ... then you can deduce ...". Axioms can be easily captured by the special case of inference rules where the condition is vacuous (you need not be able to deduce anything), in other words by inference rules of the form "You can always deduce ...".
Concerning the axioms of probability, they are not stated in a vacuum, but within an over-arching formal system. Let's take a simpler example of the axiomatization of groups:
A group is an ordered triple $(G,∘,e)$ where $G$ is a collection and $∘$ is a binary operation on $G$ and $e \in G$ such that:
$\forall x,y \in G\ ( x∘y \in G )$.
$\forall x,y,z \in G\ ( (x∘y)∘z = x∘(y∘z) )$.
$\forall x \in G\ ( x∘e = x = e∘x )$.
$\forall x \in G\ \exists y \in G\ ( x∘y = e = y∘x )$.
Notice that these axioms are more-or-less first-order sentences applied to $(G,∘,e)$. But we certainly cannot deduce a lot about groups using only the axioms, such as the fact that every element in a finite group has a finite order that divides the size of the group. There is no easy explanation of this, but intuitively it is because if you restrict yourself to working within the group axiomatization you would only be able to deduce things from the perspective of a general group, and the language you have is not even able to refer to multiple elements, iterated operations or collections of groups, not to say reason about them.
This is why group theory (and every other branch of mathematics) is investigated 'outside' the group axiomatization, in the foundational system which allows you to set up axiomatizations in the first place. The most common foundational system in modern mathematics is usually said to be ZFC set theory, but actually it is ZFC plus a fair amount of syntactic sugar to help with actual usage. The language of pure ZFC only has a single binary predicate symbol "$\in$", making it very cumbersome to express anything of interest, but if we add on-the-fly definitorial expansion then it very drastically condenses proofs and the resulting system ZFC* is a reasonably usable foundational system.
In real mathematical practice mathematicians do not even use ZFC*, but rather most mathematics is done in a fairly informal way, with the main criterion of correctness being that any sufficiently trained reader is capable of convincing himself/herself that the argument can be translated to a formal proof in ZFC*. (Actually some mathematicians do not even know the formal specifications of ZFC, but they know what they are allowed to do that set theorists tell them is translatable to ZFC.)
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1So, yes, De Morgan's laws (for sets) can be proven in the foundational system, not if you restrict yourself to say Kolmogorov's probability axioms alone. After all, his axioms use the notion of sets, and so we already need to have some foundation that axiomatizes sets. The probability axioms merely axiomatize the notion of "probability". – user21820 Jan 15 '17 at 03:19