Abstractness, Number Theory, and Logic. Flashcards
What did Neils Heinrich Abel do for polynomials?
(1820s)
Proved that there can be no formula for the quintic.
What were the two questions left in Abel’s wake?
How can we tell if a quintic is solvable?
What about higher order equations?
Evariste Gaulois?
Proved that one couldn’t trisect an angle.
Was some 17 year old.
Invented group theory.
What was Euler’s three infinities?
1) The infinitely small
2) The infinite in number (1, 2, 3..)
3) The infinite in measure (lines)
(basically number and measure)
Georg Cantor?
Came up with the men-women stadium idea.for pairings.
Came up with the Cantor zig-zag (A way to map one-dimensional space to two-dimensional space)
Figured out how much interference we could have and still recognize the image
Cantor and Dedekind
Showed how to construct all real numbers as limits of rational numbers.
Do fractions take up space on the number line?
Nope, fractions take up no space at all. A collection of aleph-not points has a magnitude of zero.
How many points are there in an interval of real numbers?
There are 2^aleph-not points, and they can collectively represent any magnitude, from [0, 1] to [-56, 123512]
How much interference can you have in order to still be able to recognize an image?
You can have no interference. That is, aleph-not points missing
What are the five axioms of arithmetic?
1) 1 is a number.
2) If n is a number, then it has a successor (n + 1)
3) You can’t have the same successor to two different numbers. (If m does not = n, then m + 1 does not = n + 1)
4) Numbers can’t go in circles (There is no natural number x such that x + 1 = 1)
5) Principle of Induction
What is fictionalism?
The idea that everything’s made up.
What is constructivism?
The philosophy that knowledge is generated from the interaction between our experiences and our ideas.
What is formalism?
The philosophy that things have no deeper meaning other than what we assign to them.
What is intuitionism?
The philosophy that truth is constructed in the mind, and that communication between mathematicians serves to tell the other what the first did to create truth.
Kurt Godel
The Incompleteness Theorem! Proved that for any logical system that includes arithmetic, there will always be statements that are neither true nor false.
