[seqfan] Re: A004249, A007516

[Maximilian Hasler]

>>i.e. in A007516 the intention is NOT to think of an empty ^ tower,
>>but of nothing written before the +1:
> This isn't mathematics, it's playing games with notation.

yes, I completely agree.
But there are hundreds of other sequences which are nothing but
"playing with notation".

>>%C A004249 For n=0, the empty product is meant.
>>(or something similar).
>
> It isn't the empty product, it's the empty tower.  The value of the
> empty tower is 1, for the same kind of reasons that ...

I know, but it seems that the seqfan archives will be the first web
page to contain the term "empty tower" in this sense (not to say that
this vocabulary did not yet exist), while "empty product" is quite
universally known.

Note also that the "tower" (i.e. power) is different from sum,
product, union, intersection ... in that the operation is
non-commutative and non-associative ;
even its (left vs right) associativity is not universally defined
(although right associativity seems the most "useful" choice).

This is relevant insofar as the "empty tower" (equal to 1) is only a
"right neutral element", if you add it on the left (at the base) it
changes the value of the whole tower to 1.
The other "empty"s (sum, product, union) are (bilateral or two-sided)
neutral elements for the given operation, which you can "insert"
anywhere in an existing product.

> the empty universal quanitifier is "true", why the empty existential
> quantifier is "false", why the empty union is the empty set, etc.

where "etc." stands for "the empty intersection is ....
the whole universe ?"  ;-)

Maximilian