@Cyracuz,
Cyracuz wrote:
Quote:Thus, according to set theory, zero is the same and also not the same as nothing.
That's because "zero" is a symbol we assign meaning to.
It can mean "nothing".
It can also mean a beginning, as in when counting the years of a person. If you count back to zero you get to the year he was born.
It can mean an end, when the gas tank of your car goes empty and the needle on the metre shows zero.
It can even mean a standard or reference point, as is the case with measuring temperature being linked to the freezing point of water. That point is zero, the boiling point 100. But we can also measure -100.
Zero is a very useful tool. Whenever it means "nothing", there is always a reference to which thing or phenomenon is being described as zero.
You keep confusing symbols with concepts: it is as a concept, not as a symbol, that zero is both different and identical to nothing. Again:
1. A set with no element is the same as a set with zero elements, by which no element is the same as zero elements.
2. A set with a zero element is not the same as a set with no element, so element zero is different from no element.
In both cases, we are not talking about the symbol of zero, but about its concept, which is, in both cases, the same concept: it just happens that the concept of zero
is two different things (we cannot say it
means two different things -- except as a pleonasm -- because we are already talking about a meaning -- a concept). If you still cannot see that, then just replace the word "zero" (or the symbol "0") by any (other) symbol -- say "$" -- so you can see that the concept remains the same:
1. A set with no element is the same as a set with $ elements, by which no element is the same as $ elements.
2. A set with a $ element is not the same as a set with no element, so element $ is different from no element.