Proof that Zero is a Natural Number
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- čas přidán 25. 05. 2024
- A step by step proof of the first Peano Postulate, that Zero is a Natural Number.
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What they told me in grade school was pretty similar. The definition of the natural numbers is all of the whole numbers and zero. Therefore, zero is a natural numbers by definition.
I thought it was the other way around. Whole numbers are the set of natural numbers along with zero.
@@blackbacon08yup, that's what I was told too. We were even given a little memory aid. We were to remember that the _WHOLE_ numbers were the ones with the _HOLE_ in them -- I.e. the zero. 😂
But I suspect that the Natural/Whole distinction really is a grade school level semantic thing and not particularly important mathematically. In fact, I'm not sure I ever heard much said about Whole numbers once I reached university. From there it was all Naturals (which would have included zero), Integers, Reals, and so on.
ADDITION: I just had a bizarre conversation about this with ChatGPT. If I could have punched it, I would've. That led me to dig around in Wikipedia, and to my surprise and delight there is a very recent, perhaps still ongoing discussion about it on the Talk page for the "Natural Number" article. It's here: en.m.wikipedia.org/wiki/Talk:Natural_number#Is_0_a_natural_number? (Careful - the question-mark at the end of that is part of the URL, but you might have to add it yourself). Whodathunk there was so much deliberation of W vs N!
negative integers are also whole numbers, whole number is just a different word for integers
@@icew0lf98 well according to the dictionary, yes. But also according to the dictionary, natural numbers do not include zero. So, I used the whole numbers term because obviously the definitions in this video are different.
@@icew0lf98 I guess the bottom line is that there are various interesting sets of numbers, and there are various labels we use to refer to them. The sets are usually pretty precisely pinned down and defined. But we humans can sometimes be a wee bit sloppy when it comes to labeling aspects. Thus we can be fairly confident with:
Some interesting sets:
{1,2,3,…}
{0}
{0,1,2,3,…}
{…,-3,-2,-1}
{…,-3,-2,-1,0}
{…,-3,-2,-1,0,1,2,3,…}
But care is needed when applying labels containing the likes of "whole", "natural", "integer", "positive", "negative", and so on.
The sets: are simply there and nothing to do with us; objective, certain, clear.
The labels: they're pretty much all us; subjective, uncertain, and sometimes as clear as mud.
As long as we don't mistake sets for labels, or vice versa; and allow for some flexibility in the mapping between them; I reckon we'll be fine. 🤓
wait, that argument seems to me to be a bit circular...
before ZFC, the DEFINITION of the natural numbers (as it was tought to me in grad school) were the peano axioms, so when building the numbers with sets, what is done is to build them SUCH THAT you can prove the postulates, therefore proving that those sets are actually natural numbers (what you said implies that the REASON for the postulates are the numbers as sets, and i disagree). This could be the case if ZFC came before Peano arithmetic
hopefully no one uses this video to disprove someone who takes the 1 as the smallest natural number, since the answer here depends on how you defined inductive set (weather it "must include 0" or "must include 1")
Around the 4 minute mark, why don’t you just assume IN(B) for an arbitrary B, and then just use the definition to derive 0€B, followed by universal intro to get the desired result? Just curious.
So we prove zero is a natural number by defining it as a natural number?
Why would you define zero as equal to the null set???
Works well.
If one defines the lexicographic order on a product of two linearly ordered sets, then the product of two finite von Neumann ordinals, given the lexicographic order, is order isomorphic to the von Neumann ordinal which is the usual product of those natural numbers,
Same thing if you use a disjoint union.
Really, it’s just how it should be?
because both is treated as False in python 😀
@drdca8263 so if the only solution to f(n)=0 is the null set, does that mean that f(0)=0 ?
@@manuel9219 this is a result of using similar notation to mean different things in different contexts.
When x is a subset of the domain of f, f(x) denotes {z | (y,z) is an element of f, and y is an element of x}
while if x is itself an element of the domain of f, f(x) denotes the unique z such that (x,z) is an element of f.
If we used different symbols for “applying a function to an element of its domain” and “taking the image under a function of a subset of its domain” there would be no ambiguity. We don’t use different notation for these because the meaning is typically clear from context.
Lost me at step 4-5🫤