Is there a notation for the last element in a set?

AI Thread Summary
In set theory, the last element of a set is not typically denoted since sets are unordered collections. The notation "max(p)" is used for the maximal element, but there is no widely recognized notation for the last element like "last(p)." The concept of a last element depends on the specific enumeration of the set rather than the set itself. In ordered contexts, such as paths in a graph, the last element can be identified, but this is not applicable to sets in general. Therefore, while there are ways to refer to the last element in ordered structures, it lacks formal recognition in standard set notation.
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Say set p = \left\{p_1, p_2,\ldots,p_n\right\}, how to notate p_n without saying p_n, since I'm not explicitly writing out p like that for a variety of reasons?
Like max(p) has the maximal element, what about last(p) or something? Is that recognized?
 
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If it is truly a set then no, that last element does not depend on the set, but on your enumeration of the set.
 
My p is a path on a graph from node 1 to 2 to 6, to etc, so here it's ordered in some sense or another...
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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