2017年9月21日 · Suppose we did start with some notion of "natural number" which we used to construct a model of the real numbers. Then even in this setting, the quoted definition is still not circular, because it's defining a new notion of "natural number" that will henceforth be used instead of the previous notion of "natural number".

2016年4月3日 · (using the inheritance property what you have said was true). In R, N should not be open since no neighborhood of maximal distance r around any natural number should have only natural numbers in it (i.e. take r = 0.5) $\endgroup$ –

2013年3月15日 · At that point, you cannot actually distinguish between the natural numbers starting at $0$ and the natural numbers starting at $1$, except for the (completely arbitrary) name for the initial number. Basically, it is the different definition for addition and multiplication which distinguishes the two choices. $\endgroup$

2018年12月4日 · Within set theory, having the natural numbers $\\mathbb{N}$ built as the minimal inductive set with the corresponding additive and multiplicative operations defined, integers $\\mathbb{Z}$ can be set...

2015年4月5日 · The positive integers are $\mathbb Z^+=\{1,2,3,\dots\}$, and it's always like that. The natural numbers have different definitions depending on the book, sometimes the natural numbers is just the postivite integers $\mathbb N=\mathbb Z^+$, but other times the natural numbers are actually the non-negative numbers $\mathbb N=\{0,1,2,\dots\}$.

Just based on Ravichandran's assessment of what the natural numbers are, the map that sends each number spoken by Sisyphus to the number spoken by his companion at the same time is obviously an isomorphism between the "Greek natural numbers" and the …

2015年6月7日 · The answer to this question is unknown, by definition, and it is known to be a natural number, also by definition. And while in general, for two unknown numbers you cannot say which one is smaller, for this specific problem we know, again by definition, that it …

2021年10月19日 · ZFC does have a notion of the "set of natural numbers" and can be definitionally extended to include such a set, as seen above. However, A is potentially wrong in that A assumes that the set $\mathbb{N}$ , defined in ZFC, has anything to do with the "actual natural numbers".

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2022年10月10日 · There occurs a pattern which I cannot describe when you keep dividing natural number by 2. It gets separated in two groups; one which the ones are ...25 and the other which ones are ...75. For exam...