Gödel's incompleteness theorem presents us with the possibility of having theorems that are neither provable nor disprovable in a given axiomatic set. Already we have the continuum hypothesis which has this property (at least in the axioms we deal with in modern mathematics today).  Are there other such mathematical statements that have been "found"?
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Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.  Phrased another way, have we proven that if a mathematical statement is true, a proof of it exists? That, therefore, anything that is true can be proven, and anything that cannot be proven is not true? Or, is there a counterexample to that statement?  If it hasn't been proven either way, do we have a strong idea one way or the other? Is it generally thought that some theorems can have no proof, or not?
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If the union of two theories is not consistent there exists a sentence $P$ for which one of them implies $P$ and the other one implies $\lnot P$.  Can someone please check if this is true or not?  I used to prove this by contradiction. I assumed for every sentence $P$, if $T_1$ implies $P$, then $T_2$ satisfies $P$ or $T_2$ doesn't imply $P$ but never $T_2$ implies $\lnot P$. So i think there exists a model of $T_2$ as a substructure of a model of $T_1$( i'm not sure if this is true... I just say it intuitively ), so I take that model of $T_1$ as a model of  $T_1 \cup T_2$. Since the union doesn't have a model so there exists a sentence $P$ such that $T_1$ implies $P$ and $T_2$ implies the $\lnot P$.
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There are some conjectures which most leading experts believe in albeit no one can prove it yet. For example: $\mathcal{P} \neq \mathcal{NP}$, the Riemann hypothesis or the Collatz conjecture.  My question is whether there is a set of such conjectures which cannot all be true, eg. is a statement such as "Either $\mathcal{P} = \mathcal{NP}$ or the Riemann hypothesis is wrong" true?
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I have a really hard time learning math when I can't find any justifications for the existence of infinity (in this case actual infinity).  Are there any justifications for the Axiom of Infinity?   Thank you very much.
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Using naive set theory and not using the cardinal number of a power set, how can you prove a set cannot contain its own power set? Is this connected to Russell's Paradox?
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What does the axiom of power set add to ZF or ZFC set theory?    More specifically, it seems that once the axiom of infinity and the axiom schema of specification are used, then all of the elements of the power set of integers, P(N), could exist.  If that is true, then wouldn't the axioms of pairing and union ensure that P(N) itself also existed?  P(N) contains many elements that are not computable sets:    Perhaps the axiom schema of specification could not be relied upon to assert the existence of any element of P(N) that is not computable (because there is no specifying formula)?  So, what is the point of the axiom of power set?  It must add something to the other axioms...  What elements of P(N) would not exist without the axiom of power set?  Bonus question:  Given the existence of P(N), could the axiom schema of specification be used to assert the existence of...:  $$\{ p \in P( \,\mathbb{N}) \, : p \text{ is a computable set} \}$$  ...where the formula for determining whether a set is computable is itself non-computable (think halting problem) and may not exist even though it is easily defined?  I'm wondering what the limitations are when it comes to formulas that can be used in conjunction with the axiom schema of specification, thinking maybe it allows formulas to be used that literally cannot exist...
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I think that the question is clear from the title:     Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$?   My feeling is that the definition of any two such sets will be circular (we can't define $B$ till we define $A$ and we can't define $A$ till we define $B$), and hence will be excluded by some axiom.  On the other hand, I can't see an inherent contradiction, though I wouldn't be surprised if some form of Russell's paradox pops up, pushed behind one more set of braces.
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This is my hypothesis and I am not sure if this is a valid one. I need a proof for this. My point is that infinite number of minimas are needed if this is valid, but I am lack of knowledge to prove it.   Any one has a proof ?   Hypotehis:  I want to show that a non-constant polynomial $P(x)$ where $x \in \mathbb{N}$ can not have infinite number of the same constant values in its range.   i.e. for some infinite set of distinct $x_i$  values, where $i \in \mathbb{N}$ and $c$ is a constant, $P(x_i) \ne c $
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This question is quite subjective, but here goes:  The axiom of choice notoriously leads to many extremely counterintuitive results, like the Hausdorff, Von Neumann, and (most famously) Banach-Tarski paradoxes. Unfortunately, the proof of these paradoxes is also notoriously complicated, and it's not at all obvious why an axiom as seemingly innocuous (and, in fact, obvious) as the axiom of choice has such strange consequences.  Do there exist any examples of a simple chain of reasoning that leads from the axiom of choice to a counterintuitive result? (Though not necessarily a result as spectacularly counterintuitive as the Banach-Tarski paradox.) One which is simple enough that someone without training in advanced mathematics could understand the general train of logic (with perhaps a few steps fudged over)? I've never see an example that lets me intuitively understand why naive intuition "goes wrong" exactly at the step where we use the axiom of choice.  (To be clear: I'm not looking for a simple counterintuitive result that follows from the axiom of choice - Banach-Tarksi would qualify for that. I'm instead looking for a counterintuitive result that follows from the axiom of choice with a (relatively) simple proof.)
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Is it possible to prove that you can't prove some theorem? For example: Prove that you can't prove the Riemann hypothesis. I have a feeling it's not possible but I want to know for sure. Thanks in advance
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prevents a universal axiomatic system for math. Is there any reason to believe that it also prevents a  for physics?    Edit:  I haven't before seen a formulation of Gödel that included time. The formulation I've seen is that any axiomatic systems capable of doing arithmetic can express statements that will be either 1) impossible to prove true or false or 2) possible to prove both true and false.  This leads to the question: Are theories of (nearly) everything, axiomatic systems capable of doing arithmetic? (Given they are able to describe a digital computer, I think it's safe to say they are.) If so, it follows that such a theory will be able to describe something that the theory will be either unable to analyse or will result in an ambiguous result. (Might this be what forces things like the Heisenberg uncertainty principle?)
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I don't know much about set theory or foundational mathematics, this question arose just out of curiosity. As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms with the axiom of choice. But the last axiom seems to be the most special out of these axioms. A lot of theorems specifically mention that they depend on the axiom of choice. So, what is so special about this axiom?  I know that a lot of funny results occur when we assume the axiom of choice, such as the Banach-Tarski paradox. However, we are assuming the other ZF axioms at the same time. So why do we blame it to the axiom of choice, not the others? To me, the axiom of regularity is less convincing than the axiom of choice (though it's probably due to my lack of understanding).
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I was reading  on the axiom of choice (AC) and it mentions that a growing number of people are moving into school of thought that considers AC unacceptable due to its lack of constructive proofs. A discussion with Mariano Suárez-Alvarez clarified that this rejection of AC only occurs when it makes sense.   This got me thinking. What are some examples of theorems in number theory that require the axiom of choice or its equivalents (ie Zorn's lemma) for its proof?  Note: Someone mentioned to me that Fermat's Last Theorem requires AC. Can someone verify this?
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Some time ago, I read something like this about the issue of &quot;a final theory&quot; in Physics: &quot;Concerning the physical laws, we have several positions as scientists  There are no fundamental physical laws. At the most elementary level, the Universe/Multiverse is essentially chaotic and anarchical. There are no such laws.  There are a continuous sequence of more and more precise theories, but there is no a final theory. Physics will be always evolving from one approximate theory to another bigger and more accurate. In the end, we will be also able to find a better theory and additional levels of complexity or reality.  There is a final theory explaining everything, and we will find if and only if: i) We are clever enough to find such a theory. ii) We make good and sophisticated enough mathematics. iii) We guess the right axioms/principles/ideas. iv) We interpretate data correctly and test the putative final theory with suitable instruments/experiments. &quot;   Supposing 3) is the right approach... Question: How could we prove the mere mathematical existence of such a theory? Wouldn't it evade the Gödel's incompleteness theorem somehow since, as a Theory of Everything, it would be explain &quot;all&quot; and though it should be mathematically self-consistent? How could a Theory of Everything be a counterexample of Gödel's theorem if it is so, or not? Note: The alledged unification of couplings in supersymmetric theories is a hint of &quot;unification&quot; of forces, but I am not sure if it counts as sufficient condition to the existence of a final theory. Complementary: Is it true that Hawking has changed his view about this question? SUMMARY: 1') Does a final theory of physics exist? The issue of existence should be tied to some of its remarkable properties (likely). 2') How could we prove its existence or disproof it and hence prove that the only path in Physics is an infinite sequence of more and more precise theories or that the Polyverse is random and/or chaotic at the most fundamental level? 3') How 1') and 2') affect to Gödel's theorems? I have always believed, since Physmatics=Physics+Mathematics (E.Zaslow, Clay Institute) is larger than the mere sum that the challenge of the final theory should likely offer so hint about how to &quot;evade&quot; some of the Gödel's theorems. Of course, this last idea is highly controversial and speculative at this point.
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I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only adequate to establish the existence of all of the natural numbers, but not also that there is an infinite set consisting of them. To do so, we must stipulate not only the Axiom of Induction, but that there also exists an inductive set (via the Axiom of Infinity).  So, why does the existence of an infinite set of the natural numbers not just follow from the existence of all of the natural numbers?
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Which is the best book on axiomatic set theory? I am interested in a book that is suitable for graduate studies and it is very mathematically rigorous.
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Excerpt from a mathematical proof that serves to prove by contradiction, using the Well Ordered Principle, that predicate P(n) is true for all nonzero integers n.     If n belongs to set C i.e. n >= c, then P(n) is false.      This implies the contrapositive,      If P(n) is true, n has to not be a part of set C i.e n &lt; c.   I know that the above implication is valid but why so? How should I know that n cannot take on 2 values simultaneously -- n being both >= c and &lt; c, thus P(n) being both true and false simultaneously? (Yes I am thinking Dialetheism, maybe even quantum logic but I know nothing other than that quantum logic defies certain aspects of propositional logic.) What is the basis behind a predicate (or any proposition) being either true or false? Or would the simultaneity be possible just that I wouldn't be dealing with a proposition to begin with?  Also, if you don't mind, please do let me know if you feel my understanding of concepts and terminologies (i.e. discrete, predicate, contrapositive, assume) are erroneous. I am extremely new to discrete mathematics and could really use the extra guidance.  Thank you.