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. $\endgroup$

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".

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 $0$ not to be a natural number: generally speaking $0$ is not natural at all. It is special in so many respects; people naturally start counting from $1$;

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 above argument follows to show uncountability and each binary representation of a real number would also represent the subset of a Natural number.

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 your notes, my guess is that your professor usually does not consider $0$ to be a natural number, and $\mathbb{N}_0$ is shorthand for $\mathbb{N}\cup\{0\}$.

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 succinctly phrased in terms of ordinals: a natural number is an ordinal that is not greater than or equal to any limit ordinal. CopyPasteIt's link has something similar ...

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 where $\omega$ is larger than every natural number, and $\omega + 1$ is larger than every natural number and larger than $\omega$.

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 that nothing can be the largest natural number because you can always add 1 to it. $\endgroup$

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 using a method by which all natural numbers have an "equal," however undefined, probability of being mapped to (here, equal simply means that no natural is given preference as ...