Comments by "kazedcat" (@kazedcat) on "Another Roof" channel.

  1. The biggest problem with abduction is confirmation bias. When we form conclusions we almost never reason to disprove our conclusion and only think of arguments that agrees with the conclusion.
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  2. For all person in the set of all humans there exist another person in the set of all humans in which person loves another person.
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  3. Not necessary. All negative numbers are additive inverse of naturals so -5 is an additive inverse of 5 which is a prime. So the prime decomposition of -20 is the prime decomposition of its additive inverse that is 20=2*2*5.
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  4.  @wesbaumguardner8829  You don't mathematics is not real. The only reason mathematics matches reality is because we choose the fundamental assumptions so that we can solve real problems. Just like simulations can be use to solve real problems because we choose to make the simulation as real as possible but simulations don't have to be real. Similarly we can build a mathematical system that does not match reality but that would be an entirely useless mathematics.
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  5. 1 is a special number that is neither prime nor composite. It is similar to zero who is neither positive nor negative.
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  6.  @ingiford175  The problem with defining -1 as prime is that it would turn all negative numbers composite. And it would break the fundamental theorem of arithmetic because you can extend any prime decomposition by multiplying it with (-1)²ⁿ for any natural number n. This means prime decomposition is no longer unique.
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  7. I think Star Trek chooses that name to describe something paradoxical. Q continuum are both in Q but also a continuum.
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  8. The assumption is that a set exist. If a set exist then you can use the subset rule to create a set with nothing in it.
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  9. There are sets not in Naturals for example sets {0,2} and {1,2} you can use this two for tagging. This set are distinct from {},{0},{0,1},{0,1,2} which are the set represented by 0,1,2 and 3. Sets {0,2} and {1,2} are not in Naturals.
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  10. What I mean is we can use this formula (a,b) = {{{a},{0,2}},{{b},{1,2}}} as definition of ordered pair. This formula always create a distinct set no matter what is in a and b.
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  11. If you don't define how heavy exactly is 1 pound you will have riot in the market.You don't like that 1 pound is too light well to bad you cannot change how heavy 1 pound is.
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  12. In foundational mathematics everything is a set or a class. So object= a specific set and property= a specific class.
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  13.  @wesbaumguardner8829  A set of anything could be apples, tractors, or unicorns. A class of any green things, or any squeezy things, or any paranormal things.
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  14.  @wesbaumguardner8829  A set of anything could be apples, tractors, or unicorns. A class of any green things, or any squeezy things, or any paranormal things.
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  15.  @wesbaumguardner8829  Mathematics only deals with sets. If you want to count unicorns then you need to encapsulate those unicorns into sets. Once something is inside a set them math does not care what is inside as long as it satisfy the rule of regularity then anything can be inside it.
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  16.  @wesbaumguardner8829  That is what mathematics is a method of abstraction. Meaning is derive from the patterns created by the relationship between abstraction. Category theory is an entire branch of mathematics describing the abstractions of the abstraction. And there is a Category for Categories. It is turtles all the way up.
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  17. It is already define with extensionality. Two set are the same set if they have the same members.
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  18.  @khiemgom  He did not use any other property except the property of extensionality. The = sign is just a symbol to represent extensionality. It is not the same as the common equality it is more specific that represents the thing on the right and the thing on the left they are identical set. There is actually other way to represent it using only symbols of first order logic without using equality but the resulting expression is very convoluted and hard to parse compared to using = to represent extensionality.
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  19. I think for vectors you need to extend the definition of ordered pairs into ordered list. Vector in 2d space is an ordered pair but in 3 dimension it becomes an ordered triples and extending an ordered pair into an ordered list takes care of that problem. Then matrices and higher order stuff can just be an ordered list of an ordered list. But this extensions from ordered pair into ordered list is a clear break from the concept of numbers.
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  20. That is the extension to extensionality he mentioned in the first video. The problem is that this axioms are define in a precise mathematical language and a lot of the specificity is lost when translated into english.
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  21. There are sets that is not a member of the set of all natural numbers. What is interesting is that we can actually use this set {0,2} and {1,2} as tags for the ordered pair structures. Since they are not natural numbers this solves the problem of finding tags that are distinct to the sets being paired. With proper construction it can be made rigorous that no matter what elements in the set the tags will always be distinct.
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