3 added 1 character in body edited Jan 15 at 0:17 Timothy 21711 silver badge99 bronze badges I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $$1 \div \infty = 0$$ outside of school but my intuition told be that $$1 \div \infty \neq 0$$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking showshown at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$$. It may have come from the Burali-Forti paradox of Naive set theory. Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $$\mathbb{R}$$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers. I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html seems to be a sign that my brain may have actually benefitted from discovering the unprovability of the axiom of choice at the age of 21. I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $$1 \div \infty = 0$$ outside of school but my intuition told be that $$1 \div \infty \neq 0$$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking show at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$$. It may have come from the Burali-Forti paradox of Naive set theory. Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $$\mathbb{R}$$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers. I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html seems to be a sign that my brain may have actually benefitted from discovering the unprovability of the axiom of choice at the age of 21. I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $$1 \div \infty = 0$$ outside of school but my intuition told be that $$1 \div \infty \neq 0$$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking shown at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$$. It may have come from the Burali-Forti paradox of Naive set theory. Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $$\mathbb{R}$$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers. I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html seems to be a sign that my brain may have actually benefitted from discovering the unprovability of the axiom of choice at the age of 21. 2 added 259 characters in body edited Jan 15 at 0:09 Timothy 21711 silver badge99 bronze badges I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $$1 \div \infty = 0$$ outside of school but my intuition told be that $$1 \div \infty \neq 0$$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking show at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$$. It may have come from the Burali-Forti paradox of Naive set theory. Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $$\mathbb{R}$$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers. I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html seems to be a sign that my brain may have actually benefitted from discovering the unprovability of the axiom of choice at the age of 21. I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $$1 \div \infty = 0$$ outside of school but my intuition told be that $$1 \div \infty \neq 0$$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking show at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$$. It may have come from the Burali-Forti paradox of Naive set theory. Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $$\mathbb{R}$$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers. I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $$1 \div \infty = 0$$ outside of school but my intuition told be that $$1 \div \infty \neq 0$$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking show at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$$. It may have come from the Burali-Forti paradox of Naive set theory. Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $$\mathbb{R}$$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers. I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result. https://www.inc.com/bill-murphy-jr/science-says-were-sending-our-kids-to-school-much-too-early-and-that-can-hurt-th.html seems to be a sign that my brain may have actually benefitted from discovering the unprovability of the axiom of choice at the age of 21. 1 answered Jan 15 at 0:01 Timothy 21711 silver badge99 bronze badges I was introduced to the concept of infinity early so I formed the intuition of the hyperreal number system. I think I also remember being taught in School that 0.999... = 1 and not understanding why that was true. I think that might have been after the time I was introduced to the concept of infinity and that might have been the reason. I was also told that $$1 \div \infty = 0$$ outside of school but my intuition told be that $$1 \div \infty \neq 0$$ which is how the hyperreal system really works. I think I may have also thought that only the notations that eventually terminate whether at finite or infinite position represent a number similar to the method of thinking show at https://www.youtube.com/watch?v=wsOXvQn3JuE as a result of being introduced to infinity early and from that assumption deduced that $$\frac{1}{6} = \frac{1}{3} \div 2 = 0.333...333 \div 2 = 0.1666...6665$$ because I had already figured out on my own how to divide by 2 in decimal. Without realizing it, my assumptions as a whole were contradictory so it can in fact be shown that $$\frac{1}{6} = 0.1666...6665 \neq \frac{1}{6}$$. It may have come from the Burali-Forti paradox of Naive set theory. Much later, I thought I finally figured out a way to construct the real numbers properly from the dyadic rationals and understood that the real number system is a different system where there are no infinite or infinitesimal numbers. I later discoved Cantor's paradox by myself. Even later, I discovered by myself that I cannot prove the axiom of choice. It turns out that without realizing it, I didn't really figure out how to construct the real numbers because I had also earlier thought of my own definition of a natural number independently which turned out to be the same as the definition on page 25 of the book A first course in real analysis which said an inductive set is a subset of $$\mathbb{R}$$ such that 0 is in the subset and for every real number in the subset, that number plus 1 is also in the subset and a natural number is a real number that every inductive set contains. Now I realize that you can construct the natural numbers as finite ordinal numbers then construct the integers then construct the dyadic rationals then construct the real numbers. I was 21 at the time I discovered on my own that I can't figure out how to prove the axiom of choice and I feel like after the time I discovered that, I got helped even more by discovering it on my own at that age than I would have by being told at a young age why it wasn't provable. I think by brain was muture enough that it could handle the change and I acutally liked breaking by old habits and learning how to adapt to think in a totally different way, unlike the Pythagoreans who hated the discovery that irrational numbers exist. I'm glad I didn't get told that at a younger age because then otherwise, my brain might have missed out on breaking its old habits at an older age and becoming even better as a result.