basic constructions:
strong axioms
further
A paradox is an example of reasoning that seems perfectly valid on first sight, but which results in a contradiction or inconsistency. For example:
The following statement is false.
The previous statement is true.
Either sentence alone seems perfectly reasonable, and could meaningfully be said in another context; but together they form a paradox. It is not just that the sentences contradict each other (in which case one or both might simply be false); but the existence of the sentences contradicts the assumption that they are meaningful at all (at least using classical logic).
Mathematical paradoxes of this form (such as Russell's paradox) are resolved by requiring stronger well-formedness rules to admissible sentences, for instance by requiring strict typing and/or guarded recursion which prevents infinite self-reference.
In contrast, paradoxes in physics may not necessarily be logical inconsistencies and tend to be resolved by rejecting unjustified assumptions and adopting more accurate laws of nature. For instance Zeno's paradox of motion is resolved by recognizing the notion of limit of a sequence; and the hole paradox is (or was) effectively a misunderstanding of the rules of differential geometry applied to spacetime.
In physics:
mathematical statements
Last revised on March 26, 2023 at 12:55:57. See the history of this page for a list of all contributions to it.