With great pleasure I am reading "Believing the axioms" by Penelope Maddy after someone linked to it here on MSE. (https://www.jstor.org/stable/2274520). However on page 495 there is a sentence I don't understand:
For example, a subset of the unit interval is called "absolute zero" if it can be covered by any countable collection of intervals.
(The 'for example' refers to 'strong smallness properties', of which being absolute zero is an example)
I don't understand this definition and due to the overwhelming number of articles on the more famous "absolute zero" from physics it is impossible to google. I think my problem is with the interpretation of the word 'any'.
It seems to me that for every point $x$ in the unit interval it is easy to construct a countable collection of intervals not covering $x$, hence showing that no non-empty set can be absolute zero. But clearly that is not what is meant. But then, what is?
The next sentence from the article does not make it any clearer:
If covering is only required when the intervals are of equal length, then the set would have Lebesgue measure zero, but would not necessarily be absolute zero.
I was hoping I could think of some criterion that sounds vaguely like 'can be covered by any countable collection of intervals of the same length' that would imply being measure zero but I failed. The next sentence from the article elaborates on the distinction between measure zero and absolute zero:
Thus Cantor's discontinuum has Lebesgue measure zero, but is not absolute zero, because it cannot be covered by countable manby intervals of length $(1/3)^n$.
'Ok', I thought, 'this gives some hint of what might be meant'. Perhaps they meant 'A set $Z$ is absolute zero if for each sequence $a_1, a_2 ,\ldots$ of numbers in $(0, 1]$ there is a collection of intervals $I_1, I_2, \ldots$ such that $I_n$ has length $a_n$ and together the intervals cover $Z$.' At least this sounds like a well defined property. EDITED IN: Due to Asaf Karagila' comment below I now know that this property actually has name and a Wikipedia page, increasing the likelihood that this indeed is what is meant. END OF EDIT
But then I thought what the analogue with 'intervals of equal length' would be and everything collapses: obviously the property 'for each number $a \in (0, 1]$ there is a countable collection of intervals, each of length $a$, that together cover $X$' is true for every subset of $[0,1]$, not just those of measure zero.
So I am back to square one. Can anyone tell me what the definition of 'absolute zero' is and what property implying 'ordinary' measure zero is intended in the second sentence?
UPDATE: after reading the wikipedia page on 'strong measure zero' I actually believe that indeed (as Asaf wrote) strong measure zero and absolute zero are the same thing. The relations of both properties to the continuum hypothesis and conjectures by Borel discussed in both articles are just too similar for them not to be the same property.
But I am still mystified what property of a given set involving collections of intervals of the same length and implying the set to have measure zero is alluded to in the second of the quoted sentences. Does anyone know?