if I google for paradoxes in math, all I find are ancient paradoxes which already have a hack or solution how to merge them out. Now I'm wondering if there are still any paradoxes in modern math, where nobody has found a way to eliminate or explain them. An example for a solved paradox would be the tortois paradox, which can be explained with infinite series.
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1The existence of paradoxes (like Russel's paradox) was the reason for the overhaul of the fundamentals of mathematics in the second half of the 19th century. Most paradoxes in modern math that I know of (like Bertrand's paradox) is most often chalked up to formulations not being as clear as they seem to be. – Arthur Jul 04 '18 at 08:56
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3It depends what you mean by "paradox". If you mean, as google defines it, "a seemingly absurd or contradictory statement or proposition which when investigated may prove to be well founded or true", then paradoxes don't really go away (with context, they lose their edge though). For example, the Banach-Tarski paradox is still fairly counter-intuitive. If you mean, a flat out contradiction (a statement such that both it and its negation are believed to be true), then no, we're fresh out. :-) – Theo Bendit Jul 04 '18 at 09:01
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2You can see an overview into Paradoxes and Contemporary Logic : a glance at present-day investigations as well as Logical Paradoxes : A Contemporary Twist. – Mauro ALLEGRANZA Jul 04 '18 at 09:02
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Regarding the tortoise, we recently had this question. – joriki Jul 04 '18 at 11:06
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1The problem is that there is not agreement about what it means to eliminate a paradox. For example, we often look at formal systems, and think that a consistent formal system will not have paradoxes. But does that actually eliminate the paradox from our unformalized natural-language mathematics? – Carl Mummert Jul 04 '18 at 12:28
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1I think this would fit better on the philosophy StackExchange site. – Carl Mummert Jul 04 '18 at 12:36
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A paradox is defined in the Cambridge Dictionary as a situation or statement that seems impossible or is difficult to understand. There are plenty of these here: https://math.stackexchange.com/questions/139699/what-are-some-examples-of-a-mathematical-result-being-counterintuitive and I'm going to vote this question a duplicate of that. – it's a hire car baby Jul 04 '18 at 16:16
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Possible duplicate of What are some examples of a mathematical result being counterintuitive? – it's a hire car baby Jul 04 '18 at 16:17
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I assume you mean contradiction by the term paradox. Modern mathematics is based on formal axiomatic method. Particularly ordinary mathematics can be derived from ZFC set theory. No such paradox so far has been derived within ZFC. By Gödel's Incompleteness Theorems, in fact it is not possible to know within ZFC whether or not ZFC contains any contradictions. This does not mean that ZFC is inconsistent. It just means that "if it is consistent, it cannot be proved within ZFC".
acevik
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I think it is reasonable to add to the answer by acevik that not all mathematicians accept the C, i.e., the axiom of choice (AC) in ZFC set theory (for example constructivists, finitists, etc.). In this case, the "Banach-Tarski Paradox" is considered a paradox. Essentially, the answer depends on the own point of view on mathematics. An interesting book in this regard is among others e.g. Saunders Mac Lane's "Mathematics. Form and Function". – InfinitelyInquisitive Jun 26 '19 at 13:53