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For me, facts like the independence of the continuum hypotheses from ZFC cast a doubt on the "law of the excluded middle". (In this context, the doubt is that there might be no "final set theory" such that every proposition about sets is either true or false.) Now I'm looking for simpler (mathematical) examples that are easier to understand, but still cast a similar doubt on the "law of the excluded middle" (or on another law of the three classical laws of thought). Right now, I think that recursion theory (also known as computability theory) might teach me the sort of examples I'm looking for (at least I can appreciate the difference between primitive recursive functions and $\mu$-recursive functions).

Here is an example closely related to "computability theory" that I "nearly" understand:

For a sequence $(a_n)_{n\in \mathbb N}$ of natural numbers, the set of all natural numbers occurring in the sequence $U_a:=\cup_{n\in \mathbb N}\{a_n\}$ is a subset of the natural numbers. Each subset $U$ of the natural numbers gives rise to a unique sequence $(a_n^U)_{n\in \mathbb N}$ with $a_n^U < a_{n+1}^U$ and $U=\cup_{n\in \mathbb N}\{a_n^U\}$. Two subsets $U$ and $V$ of the natural numbers are identical if and only if the sequences $(a_n^U)_{n\in \mathbb N}$ and $(a_n^V)_{n\in \mathbb N}$ are identical. Comparing two sequences for equality is easy, at least when the sequences are given explicitly in a computable form. However, it may happen that a computable sequence $(a_n)_{n\in \mathbb N}$ gives rise to a set $U_a$ for which the sequence $(a_n^{U_a})_{n\in \mathbb N}$ is not computable. Even worse, there are computable sequences $(a_n)_{n\in \mathbb N}$ and $(b_n)_{n\in \mathbb N}$ for which there exists no computable way to determine whether $U_a$ and $U_b$ are identical.

It seems to me that all references to "computable" in this example can be made exact by interpreting them as "computable by a $\mu$-recursive function", except the statement "there exists no computable way".

  • Is there a way to also give the statement "there exists no computable way" a well defined meaning in the context of recursion theory?
  • Can the example be further simplified? Is the statement "there exists no computable way" necessary at all for understanding the "undecidability properties" that allow the "law of the excluded middle" to not apply (in certain mathematical contexts)?
  • Are there simpler examples using less infinite objects and assumptions, for examples just working with sequences of 0 and 1 without making the rest of the construction significantly more complicated?
  • Are there any questionable assumptions or axioms used by the example above, except the assumption that the union $\cup_{n\in \mathbb N}\{a_n\}$ always exists?

Edit The comments below show that it is quite difficult (for me) to explain what I actually would like to know. Especially, there was the suggestion to change the title of the question. The problem is that the title is actually quite appropriate for my question, but that mentioning of the "laws of thought" seems to divert the attention of the readers completely from what I actually want to know. There is also a limit to how much I want to change an existing question by editing. Trying to clarify the question is fine, but if it should be necessary to turn it into a completely different question I prefer to ask another question instead.

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I don't see that this casts any doubt on the law of identity. Difficulty in recognizing that two descriptions actually describe the same object is hardly the same as actual 'fuzziness' of identity. –  Brian M. Scott May 6 '12 at 23:16
There is a concept of "equivalence", which is coarser than that of identity. When we talk about "representations of objects", we are dealing with equivalence, so there is no problem with the fact that two different things may represent the same object. In any case, this would be in the "opposite direction" as the law of identity, which saysthat identical things are "equal", not that two "equal/equivalent" things must be identical. I don't see the connection with computability, though. –  Arturo Magidin May 6 '12 at 23:18
"Even worse, there are computable sequences (an)n∈ℕ and (bn)n∈ℕ for which there exists no computable way to determine whether Ua and Ub are identical." In some sense this is not quite true. Because they are computable sequence they have index $i$ and $j$. The answer to whether the sequence given by $i$ and $j$ are equal is a finite piece of information - yes or no. However, if you want a uniform procedure such that given any index $i$ and $j$ to determine if the sequence is the same, then no such computable function exists. –  William May 6 '12 at 23:34
But the statement "There are exactly $n$ distinct numbers $\zeta$ such that $\zeta^k=1$*" is a statement which is true or false in different models of field theory where $n$ can vary anywhere between $2$ to $2^{\aleph_0}$. *How is this any different than the statement "There are only $\aleph_\alpha$ many sets of natural numbers." other than a psychological pan? Think about this, please do, not for ten minutes. For a few days. I gave this thought about two years in which I studied set theory, I don't expect anyone who's not studying set theory in full to get an answer in ten minutes. –  Asaf Karagila May 8 '12 at 23:25
If your question is "Are there (simpler) examples which (more clearly) cast doubt on one of the three classical laws of thought," then why is your title "Equivalence of sequences and subsets of natural numbers"? Wouldn't a more congruent title be something like, "Examples casting doubt on the laws of thought"? –  Gerry Myerson May 9 '12 at 0:02

1 Answer 1

Here's a doubt we can have about the LEM: First you'll need to have proven Goedel's incompleteness theorem: for Peano Arithmetic, $PA$, there is an arithmetical sentence $S$ such that $PA$ does not prove $S$ and $PA$ does not prove $\lnot S$. Since $PA$ is a first-order theory of classical logic, for any sentence $X$, $PA$ proves $X \lor \lnot X$. In particular then, $PA$ proves $S \lor \lnot S$. Yet $PA$ proves neither disjunct. So the meaning of $\lor$ in classical logic seems to be just wrong in some cases: we have proven a disjunction yet we can prove neither disjunct. Then first-order theories lack the so-called disjunction property.

In contrast, if we take the same axioms as $PA$ yet use intuitionistic first-order logic, where we take classical FOL and discard (one of the axioms equivalent to) LEM, then we do have the disjunction property: Anytime this system $HA$ proves $A \lor B$, then either $HA$ proves $A$ or $HA$ proves $B$. So there's a dilemma we face: either have classical logic and have $\lor$ behave deviantly, or use a non-classical logic, like intuitionism, have the disjunction property, at the cost of (among other things) a more complicated semantics for the logic. Granted this is also an example also using an independent statement, but I would submit this is less "advanced" than CH, and maybe it's a helpful way to rephrase what your worry was about LEM.

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+1 This seems quite helpful to help me better understand my doubts about LEM. (I still have to think a bit about it.) My problem with using Goedel's incompleteness theorem directly was not that I didn't understand its proof, but that every time I would have proved an axiom system incomplete, another axiom could be added to the axiom system rendering my proof obsolete. So I was looking for an example that didn't depend too much on the used axiom system. Your answer avoids this problem by explaining why FOL doesn't enforce LEM for a huge class of potential axiom systems. –  Thomas Klimpel May 13 '12 at 12:11

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