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I think that the matter of the paradox is that it causes explosion and trivializes the formal system, and if this explosion can be prevented, there is no problem.

I'm an unprofessional person but come up with an idea.

At first,

1.discard the general law of excluded middle and disjunctive syllogism.

Next, there are two kind of propositions. One is which must not contradict itself (namely, $P\wedge \lnot P$ can not be admitted), the other is which can contradict itself. An example of former is rigorous scientific claim and examples of latter are purely formal artificial statements, or sentences in ordinary contexts such as "I like him and I dislike him.".

Then,

2.admit law of excluded middle and disjunctive syllogism only for proposition of former kind.

3.the propositions "$X\in X, X\not\in X$" for the Russell set $X$ are of latter kind( namely, are artificial ones and don't appear in natural mathematical context), so doesn't cause explosion.

Now, I think that

4.in spite of both $X\in X\Rightarrow X\notin X$ and $X\notin X\Rightarrow X\in X$ are true, explosion doesn't occur because none of $X\in X$, $X\notin X$ and $X\in X \lor X\notin X$ is true (unless there are some extra axioms which prove them).

5.we can do usual mathematics because of 2.

Does this idea work?

Repply

How do you rigorously identify "artificial" statements?

->I define natural statements individually and identify artificial ones to the others. For example, one can assume statements which only relevant to sets in Gödel's constructible universe to be natural ones.

you may be trying to (re?)invent some form of paraconsistent logic.

->I don't know paraconsistent logic technically but I think that if we try to paste more than one scientific theories together or treat natural language, it is natural that there will be some contradictions and we have to treat them in some way. For this purpose, I discarded the full-strength explosion law.

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    $\begingroup$ How do you rigorously identify "artificial" statements? $\endgroup$ – Noah Schweber Mar 18 '16 at 15:53
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    $\begingroup$ It's already been solved - the solution is ZFC $\endgroup$ – Tom Collinge Mar 18 '16 at 16:02
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    $\begingroup$ The kind of "monster barring" strategy you describe has a long history, sometimes ultimately not so successful (Russell/Whitehead) and sometimes extremely successful (Zermelo-Frankel Set Theory). $\endgroup$ – André Nicolas Mar 18 '16 at 16:09
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    $\begingroup$ @AndréNicolas: I think it is a little early to comment on the ultimate success of Russell & Whitehead's ideas (and I know many advocates of various kinds of type theory who would object much more strongly than that $\ddot{\smile}$). $\endgroup$ – Rob Arthan Mar 18 '16 at 22:41
  • $\begingroup$ It sounds like you may be trying to (re?)invent some form of paraconsistent logic. $\endgroup$ – Henning Makholm Mar 19 '16 at 15:43
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Your point (3) makes no sense at all. What precisely are you forbidding? The Russel set is just $R = \{ x : x \notin x \}$. There is absolutely no mention of $R \in R$ in the definition of $R$ itself. You cannot say that you ban the definition of $R$ because it subsequently leads to a contradiction. Otherwise there is a much simpler alternative: Ban any statement that subsequently leads to a contradiction! (*)

For a formal system to work, you must precisely specify the rules governing the syntax of the statements and the deductive steps in proofs. Specifically, you would have to have a deduction rule that specifies exactly when you are allowed to construct (equivalently state the existence of) some object that satisfies some properties. If the formal system is a kind of set theory, we usually have an axiom of comprehension (compare the comprehension axiom of ZF and of NFU!), which gives rise to existential statements that are externally interpreted with the intended meaning of stating the existence of certain collections called sets. $\def\eq{\Leftrightarrow}$

Just to expand on my first paragraph, you cannot say that you disallow constructing a collection $S = \{ x : φ(x) \}$ whenever it would lead to $S \in S \eq \neg S \in S$, simply because that latter is true if and only if the collection is contradictory! That means that you are doing no better than (*), which is of course useless because we cannot do mathematics without any rule that prevents writing down contradictory statements before we discover the contradiction. That is in the first place the entire goal of formalization, which is to reduce reasoning to sound rules!

If you don't believe this, consider:

Let $S = \{ (x,y) : (x,1) \in x \eq y=1 \}$.

Let $T = \{ (x,y) : (x,1) \notin S \eq y=1 \}$.

Then $(T,1) \in T \equiv ( (T,1) \notin S \eq 1=1 ) \equiv (T,1) \notin S \equiv \neg ( (T,1) \in T \eq 1=1 )$

$\quad \equiv \neg (T,1) \in T$.

Contradiction!!!

Which is not allowed, $S$ or $T$, and why?

So any rule that you want to concoct for an alternative set theory (of which there are tons already) would have to precisely specify at the point of set construction whether or not some set can be constructed, without referring to any future deduction of contradiction. Note how ZFC does it by restricting comprehension to only subsets of previously defined sets, while NFU does it by restricting the kinds of formulae allowed in defining sets. Both do not require knowing whether there would be a contradiction later, and so if they are consistent then we would be happy.

In contrast, if you enforce consistency by saying that we can write anything as long as it remains consistent with what we have written down, then we are in big trouble. Firstly, it means that we can never know what is allowed to be written down! Secondly, even if we did know what is allowed (say an omniscient being told us), we would under mild conditions be able to write down a certain sequence of allowed statements leading to a certain conclusion, and write down another sequence of allowed statements leading to the opposite conclusion! (This is due to Godel's incompleteness theorem.) To say that both of these are very very bad would be a severe understatement.

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  • $\begingroup$ @user305307: To understand better what is going on, I suggest you learn the foundations of logic (see math.stackexchange.com/a/1684208/21820) and then continue with Peter Smith's Godel Without Tears. And then maybe in the future you will come up with another way to get around those pesky prickly paradoxes that is better than existing foundational systems! $\endgroup$ – user21820 Mar 19 '16 at 15:11
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To quote from your recent edit:

I define natural statements individually and identify artificial ones to the others.

That's the problem. First of all, as long as you keep approaching things like this, you won't be able to write down a formal description of your system. Maybe that doesn't bother you, but one of the reasons we like formal descriptions is that they prevent confusion:

  • Are you confident that you will always be able to decide whether a sentence is artificial or natural, and that you will never change your mind?

  • Are you confident that others will be able to do the same, and will make the same decisions you do?

If your answer to the latter question is no, then what you've described can't really be used as a foundation for everyone's mathematics - and if your answer to the former is no, then this isn't even a firm ground for your own mathematics!

Now, maybe your answers to each question is yes: that there are no "dubiously natural" sentences, and that mathematicians will be able to agree on what is natural and what isn't. This is a kind of optimism similar to that in an argument proposed by Kalmar, who focused on truth rather than naturalness (see https://mathoverflow.net/questions/233234/are-there-finitistic-nonrecursive-functions-assuming-churchs-thesis-is-false); this is a type of optimism I (and, I think, the majority of the mathematical community) strongly disagree with.


Basically, in order to be happy with a system as a foundation for mathematics, at the very least I need to be confident that the system will unambiguously tell me when a proof is correct or not. When we bring subjective judgment into the mix, this goes out the window: if I have a proof which uses LEM, I now need to know that I used it only for "natural" sentences, and without a precise definition of naturality I can't know whether or not my proof is correct.

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How do you rigorously identify "artificial" statements? That was the intuitionists' critics against the Hilbert foundation program. Hilbert meant that his system managed to define mathematical objects while Brouwer and the other intuitionists' meant that Hilbert's formulas only where strings of signs.

I think that this question belongs to the gray zone between mathematics and psychology. The psychology of humans!

There is nothing wrong with statements as $\: x\notin x\:$ or $\:x\in x\:$, after all '$\in$' is just a relation among others. The problems arises when trying to define some mathematical objects, as $\{x|p(x)\}$. For a long time everyone toke for granted that there was a one-to-one correspondence between predicates and classes, but Russell's paradox show there is no such correspondence.

With a similar construction one can show that there is no one-to-one correspondence between a set and it's powerset.

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  • $\begingroup$ Do you mean "meant" rather than "ment"? Indeed we ought to add philosophy to the mix! For intuitionists mathematics is about mental constructions, whereas for platonists mathematics is about a reality, whereas for formalists mathematics is just a symbol-pushing game. Almost everyone has at least a bit of all three! I'd also clarify that although there is no internal bijection between a set and its powerset in ZFC, there might be an external one, because there is a countable model of ZFC, in which of course the set of real numbers would be externally countable! $\endgroup$ – user21820 Mar 20 '16 at 8:02
  • $\begingroup$ @user21820, thanks for the correction! $\endgroup$ – Lehs Mar 21 '16 at 3:30

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