⇒x=y
Regularity: (also called foundation) is used to prevent sets that don’t eventually resolve to elements, like a₀∋a₁∋a₂∋a₃∋a₄∋a₅∋⋯
And thus also cases like x∈x.
It does this by saying that when you order it this way, it must eventually run out of things. So at the end if these sequences there’s always an element that contains none of the other elements. Unless there are no elements in the sequence.
Or, every set contains an element disjoint from the set.
∀x x=⦰ ⋁ (∃y∈x y∩x=⦰)
(Schema of) Specification: (or Separation)
This defines the behaviour that you can filter out a set. Or, for every class of elements, there is a set that can be made by only including the elements that are also in the class.
∀z∀*a∃y ∀x x∈y ⇒ x∈z ⋀ φ(*a, x)
(One of these for each φ)
I am using the asterisk to denote a₀, a₁, … Up to some a sub n.
Pairing:
For any two elements, there is a set contaning those elements. (You can include an “exactly those elements” instead, but separation allows you to do that too.)
∀x∀y ∃z x∈z ⋀ y∈z
Union:
For a set, there is a set such that all the elements of elements of the first set are elements of the second set. Similar to above you may need to include an exactly if you are not working with separation.
∀x∃y∀z (∃w w∈x ⋀ z∈w)⇒z∈ww
Replacement:
For any function class, there is a map from a set to the image of the set under that class. You can again use separation to get the cleaner version.
∀w∀*a∃x ∃y (y∈w ⋀ ∃z F(*a, z, x))⇒y∈x)
Infinity:
There is a nonempty set that is closed under succession. (Which is a structure we want for the natural numbers.)
∃x ⦰∈x ⋀ ∀y y∈x ⇒ S(y)∈x
Where S is the successor function class that maps x to ∪{x,{x}}
Power set:
For every set there is a set of all of its subsets.
∀x∃y ∀z(∀w w∈z ⇒ w∈x)⇒z∈y
Axiom of Choice. Well Ordering Principle, Zorn’s Lemma.
Same thing. Number of ways to state it. This one is controversial sometimes, but very useful. It claims that you can choose one item out of each set in a set of sets, and get a valid set. The is not something necessarily possible with the rules above.
…
Real life has called but I will continue with the rest.
Alright we’re doing a couple really fast while I wait for a bus!
Two drawings of the Peterson Graph being isomorphic to each other.
Triangle numbers inside a right triangle with squares on each side length, in reference to the Pythagorean theorem.
The top right one has one of the definitions for Euler’s number, being the limit of that sum. The bottom one is a multiplication table up to 4 of the natural numbers. The third one has an appolonian gasket approximating the shape, with the area of a circle in the middle.
The squares of Fibonacci numbers denoting the areas of the squares used to construct the Fibonacci spiral, and Roman Numerals from 1 to 10, inclusive.
Each layer is the running sum of everything in the previous layer up until there. At the top is the omega combinator. Then there is a shift in bases.
Ahh! I recognise some more but I have to go again!
These sums are the same in a Sudoku.
The Cayley graph of D₃.
Filling a square with the larget remaining power of φ.
Koch Curve and Serpinski Triangle along with their Hausdorf dimension. The one at the top isn’t obvious to me but possibly the Cantor Set due to the dimension.
(P⇒Q)⋁(Q⇒R) is a true fact. If you use A⋁￢B to mean B⇒A, which is a common definition of ⇒, then this becomes:
￢P⋁Q⋁￢Q⋁R
And that’s true by LEM.
This is not necessarily true without LEM.
Packing problem at the top. Subtractive colour mixing at the bottom. And a tesseract in the middle.
Klein Bottle at the bottom. I am unsure what the top one is. Two pentagons overlapped showing (similar?) triangles formed by lines between the intersections but I have not seen it before. What is it?
The automorphisms of quaternions are isomorphic to the symmetry group of four elements. This is a neat statement and I like it.
Fibb sequence in binary, going up, with squares on the left.
A right triangle can be constructed (from a given area) with any perimeter more than the square root of twice itself. In this case perimeter 7 has an area of square root 2. I am unsure if this has any additional properties.
A sum of fractions is up there too.
A square with 1-9 with all the horizontal, vertical and both main diagonals add up to 15.
Powers of 3 and something else on the zig zag.
And the sequence of numbers at the top are busy beaver numbers. And at the bottom there are a bunch of ordinals. I thought I had taken pictures of them but apparently not and now I’m too lazy.
So… questions: zig zag and pentagons?
those two are an illustration that prime numbers larger than 3 are 6k±1, and a 9/2 enneagram with the Xen inside it
(via mathhombre)
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