Reading part 5 of Category Theory Illustrated by Jencel P. on stream.

• Previous reading is 20251211220616
• Logic is also category theory
• Logic is a theory which only needs to be consistent with itself, and tries to be true for any universe that is possible to exist and is not affected by observations.
• I know from Godel's incompleteness theorems (see Math's Fundamental Flaw by Veritasium that any consistent formal system of math cannot prove its own consistency.
  • Also from this video, I know that consistency means that it does not contradict itself.
• Logic organizes its rules into logical systems, also known as formal systems.
• Set theory is just one single primitive added to the standard axioms of logic. Category theory is the same, but with a different primitive added.
• It's weird to me that they feel the need to point out the equivalence of primary and composite propositions. From my perspective, if you define an underlying set as being created from some primitive value and some operator, then you should always be able to add a primitive value to a composite. For example, if I define the natural numbers with the primitive values of 0 and 1 with the addition operation, 5 is just (1 + 1 + 1 + 1 + 1). And so, if I want to add 1 to 5, that should always work because we know that 5 is actually just five ones added together. So, it's not clear to me when this equivalence would ever be untrue for anything.
• The "Modus Ponens" example doesn't make any sense to me. They have (A->B) ^ A -> B and I don't understand what A and B could possibly be in a way that would satisfy this statement.
  • Maybe they meant (human(socrates) ∧ ∀x, human(x) ⟹ mortal(x)) ⟹ mortal(socrates)
  • Perhaps they should have used more letters here, like (E->F) ^ A -> B
  • We can also use a different example without changing the letters:
    • A - I am rich
    • B - I can buy an airplane
    • D - If I am rich, then I can buy an airplane
    • C - I am rich, and if i am rich I can buy an airplane
• The "Modus Ponens" is always true, and propositions that are always true are known as tautologies.
• Propositions that are always false are known as contradictions.
• Each tautology can be turned into a contradiction by adding a "not" and vice versa.
• An axiom schema is like the structure of an axiom. For example, "Modus Ponens" is the axiom schema and "Modus Ponens" with A and B replaced with specific propositions is an axiom.
• Chat explains that one of the diagrams shows that P → Q can be replaced with ¬P ∨ Q.
  • According to wikipedia, P → Q is known as the material conditional operator and can be read as "P implies Q" or "if P, then Q". (see: Material implication (rule of inference) and Material conditional).
  • This is true because P → Q is known as the material conditional operator, which asks "under which conditions is this statement true?". The truth table of asking this question is the same as the truth table of ¬P ∨ Q.
  • You can also think of the material conditional as asking "when can this statement be disproven?". It can only be disproven when P is true and Q is false.