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) …

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 …

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 …

2015年1月8日 · I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it. Pros of considering …

In fact it has the same cardinality as Reals as in some sense we are trying to count all the possible binary numbers in the above argument. If we look at Reals in the binary form the …

There is no general consensus as to whether $0$ is a natural number. So, some authors adopt different conventions to describe the set of naturals with zero or without zero. Without seeing …

2019年6月20日 · The set of natural numbers cannot be defined. However, the property of being a natural number can still be defined, but one needs to use a different definition that can be most …

2015年10月16日 · But then we can add an element defined to be larger than every element in this set. Let's call this element $\omega + 1$. So $\{1, 2, 3, \dots, \omega, \omega + 1 \}$ is the set …

2018年1月12日 · Once you do that, you may well find that what "the natural numbers is infinite" means "there is no largest natural number" to you -- and then it's a perfectly good proof to say …

2010年12月18日 · By partitioning the reals into non-measurable sets, one can devise a way to relate a real selected uniformly at random from an interval such as [0, 1) to a natural number …