There's a thing in maths where you want to be able to define everything as rigorously as possible. If you'll try to think how go define what is the number 1, you will probably have some trouble, since you'll think "it's just 1".
Set theory is, well, maths with sets. A set is just a collection, and a set can be empty. At some point in history, someone figured a way to use empty sets and sets of sets to properly define the natural numbers (ie 0, 1, 2, and so on). The picture shows a graphical representation of how that system would define the number 8.
This is the sort of thing a normal person thinks "I just know what 1 is", but a mathematician wants do define everything as specifically as possible, which can lead to baffling constructions (at first glance) like this one.
Depends on the context really. 0 is or isn't a natural number depending on whether or not you need it to be. For example, set theory's definition of natural numbers only works if you say that the empty set is 0, and other fields need 0 to not be part of natural numbers for them to work.
Really most definitions allow for 0 to be natural, but there are a few that were made specifically to exclude it, just like how the common definition for prime numbers specifically exclude 1.
ah, that's interesting. so it's basically the same as division by zero (or some other fundamental rule, i forgot which specifically) being undefined but in some cases it has a definition for simplicity sake?
I read this whole thred. I understed pritty much everything (I thing so), and I come to deduction that this shit is realy fucked up. AND I waste a ton of time reading this.
same, but it's interesting nonetheless to see how down bad can someone be to define something in a way even someone who hasn't grown in society (or just humanity) could possibly understand 🤣
No, the standard way is by using the Peano axioms. Which defines zero as a natural number, and all other numbers as iterations of the successor function on 0. along with other axioms to allow induction etc. The one presented in the meme is the formulation of the natural numbers by Von Neumann, which isn't the standard way.
The definitions are reminiscent of one another, but formally speaking theyre not the same. In the Peano axioms not every element needs to be a set. There is a thing called an alphabet and letters in the alphabet (these terms are formally defined in set theory). 0 is a character in the alphabet, so dont think of it as a number yet just think of it as a symbol/character/letter. And then the set of natural numbers whose elements are called natural numbers, are defined roughly as follows, the symbol 0 is in this set. For every element x in this set, S(x) is also in the set. Another axiom is that 0 is not the successor of any element in the set. The other axioms define addition multiplication and induction. In this form the symbols "1,2,3,..." is not in this set. In order for that you need to define a decimal notation for the natural numbers, or you can define a binary or other notation if you want. But the main point is that notice how 0 isnt defined as a set nor the other natural numbers, the natural numbers are strings of letters in the alphabet, and not sets. Although it is true that there is a reminiscent structure to this definition and the Von Neumann one, they're not the same formally speaking. And both these definitions are reminiscent to cavemen carving lines on the walls of their cave for counting, abstractly you should think of natural numbers as this carving thing, rather than the decimal form 1,20654,23 and so on because this form obscures the structure of natural numbers. We only use it for brevity.
It is the standard way in set theory (it is the way that can be extended to transfinite ordinals). Peano axioms (I will assume you mean the 2nd order axioms) define arithmetic not natural numbers : In set theory many sets satisfy theses axioms, Von Neumann naturals is one of them (the "standard" model of arithmetic). In Peano arithmetic (the theory) everything is a natural number so the definition of a natural number would simply be : "A natural number is anything".
Ah... set theory.
PURE SET THEORY!
okay this is the worse comment i made currently
Glad to see you’re open to worse comments.
was not expecting to get upvoted when it was basically just a copy of the previous comment but with a slight change
i barely understand math and still got the joke my brain is big and smart
what does it mean tho
There's a thing in maths where you want to be able to define everything as rigorously as possible. If you'll try to think how go define what is the number 1, you will probably have some trouble, since you'll think "it's just 1".
Set theory is, well, maths with sets. A set is just a collection, and a set can be empty. At some point in history, someone figured a way to use empty sets and sets of sets to properly define the natural numbers (ie 0, 1, 2, and so on). The picture shows a graphical representation of how that system would define the number 8.
This is the sort of thing a normal person thinks "I just know what 1 is", but a mathematician wants do define everything as specifically as possible, which can lead to baffling constructions (at first glance) like this one.
Cientists Will do ANYTHING instead of finding a cure for being drunk 😭😭🥀🥀
cientists. also there are hangover pills yk
Cyentists
Isn't being drunk the whole point of drinking alcohol though?
well yes but i imagine they would want a drunkenness killswitch or something. or theyre referring to hangovers
there are pretty good preventative measures for it though
100ists? Are they racists against the number 100?
set theory description of 8
A natural number in Set Theory is the set of all national numbers below it.
0 has no natural numbers below it so it's the empty set (in the image represented by an empty box)
1 has only 0 below it so it's the set containing only 0 (the empty set)
2 has both 0 and 1 below it so it's the set containing 0 (the empty set) and 1 (the set with only the empty set)
Continue this and it becomes like exponentially more complex, and 8 is the system of nested boxes shown in the image
so 0 is a natural number in set theory?
Depends on the context really. 0 is or isn't a natural number depending on whether or not you need it to be. For example, set theory's definition of natural numbers only works if you say that the empty set is 0, and other fields need 0 to not be part of natural numbers for them to work.
Really most definitions allow for 0 to be natural, but there are a few that were made specifically to exclude it, just like how the common definition for prime numbers specifically exclude 1.
ah, that's interesting. so it's basically the same as division by zero (or some other fundamental rule, i forgot which specifically) being undefined but in some cases it has a definition for simplicity sake?
No division by 0 is undefined because it can be positive or negative, but yeah that's the right idea.
I read this whole thred. I understed pritty much everything (I thing so), and I come to deduction that this shit is realy fucked up. AND I waste a ton of time reading this.
same, but it's interesting nonetheless to see how down bad can someone be to define something in a way even someone who hasn't grown in society (or just humanity) could possibly understand 🤣
Yea
These days almost everyone considers it a natural number. In the past there were people who did, and people who didn't.
idk why but my brain really likes this concept
8 different sizes of rectangles I guess
:pensive:
You might wanna learn about axioms.
Is this loss?
r/peterimaginestheloss
r/AlreadyHere
That's where we are
I got confused for a moment and didn't see the subreddit... o(--(
...no
In set theory, natural numbers are defined as follows.
0 is the empty set.
A natural number N is defined as the set that contains every natural number under N.
i dont think this is the standard way its defined but one of the alternatives
It is the standard way. In any case set theorists dont spend their time doing arithmetic with sets, so you could say there is no standard way.
No, the standard way is by using the Peano axioms. Which defines zero as a natural number, and all other numbers as iterations of the successor function on 0. along with other axioms to allow induction etc. The one presented in the meme is the formulation of the natural numbers by Von Neumann, which isn't the standard way.
Isn't that the exact description I gave? Or did I mess up somewhere?
0 = {}
S(x) = x + {x}
while both define the naturals, the Peano Axioms and the Von Neumann Ordinals do it by different methods
this case is the Von Neumann Ordinals
The definitions are reminiscent of one another, but formally speaking theyre not the same. In the Peano axioms not every element needs to be a set. There is a thing called an alphabet and letters in the alphabet (these terms are formally defined in set theory). 0 is a character in the alphabet, so dont think of it as a number yet just think of it as a symbol/character/letter. And then the set of natural numbers whose elements are called natural numbers, are defined roughly as follows, the symbol 0 is in this set. For every element x in this set, S(x) is also in the set. Another axiom is that 0 is not the successor of any element in the set. The other axioms define addition multiplication and induction. In this form the symbols "1,2,3,..." is not in this set. In order for that you need to define a decimal notation for the natural numbers, or you can define a binary or other notation if you want. But the main point is that notice how 0 isnt defined as a set nor the other natural numbers, the natural numbers are strings of letters in the alphabet, and not sets. Although it is true that there is a reminiscent structure to this definition and the Von Neumann one, they're not the same formally speaking. And both these definitions are reminiscent to cavemen carving lines on the walls of their cave for counting, abstractly you should think of natural numbers as this carving thing, rather than the decimal form 1,20654,23 and so on because this form obscures the structure of natural numbers. We only use it for brevity.
It is the standard way in set theory (it is the way that can be extended to transfinite ordinals). Peano axioms (I will assume you mean the 2nd order axioms) define arithmetic not natural numbers : In set theory many sets satisfy theses axioms, Von Neumann naturals is one of them (the "standard" model of arithmetic). In Peano arithmetic (the theory) everything is a natural number so the definition of a natural number would simply be : "A natural number is anything".
But the standard axioms of mathematics are ZFC, not Peano's. And in ZFC, the most common way to define the naturals is like that.
(Although Von Neumann's formulation is nothing more than an instantiation of Peano's by exclusively using sets.)
Thought ts was r/peterexplainsthejoke for a second
Same!!
https://preview.redd.it/r063szqq395g1.jpeg?width=844&format=pjpg&auto=webp&s=e759f6c7e02128c53e6a10cf68290438ad3e26d3
yesterday i woke up sucking a lemon
Kid... A?
Von Neumann would be proud
i thought this was peterexplainthejoke
1000
on that math-pilled grindset
i love set theory :3
The humble Arabic numeral ; 8
What did 0 say to 8? Nice belt!
r/unexpectedtermial
how the fuck do i count this
what in the mcfuck is option 3
coaxed into animation frames
r/suddenlyradiohead
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My kid : (lambda fx.(f(f(f(f(f(f(f(f(x))))))))))
Visualizing plus times plus ahh moment.