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yuuki ゆうき(金野裕希)

in the large picture,
• the size of a number
is proportional to
• how many numbers it can prove to be consistent.

the largest number proves the consistency of all numbers, including itself.
by Gödel's theorem, it is a "contradiction".

yuuki ゆうき(金野裕希)

by Gödel's second incompleteness theorem, the axioms "finite numbers exist" can't prove their own consistency.
• Fin ⊬ Con(Fin)

now if we add "ℵ₀ exists", then we can prove the consistency of finite numbers.
• Fin + Inf ⊢ Con(Fin)

this shows Fin + Inf > Fin.

Replied to yuuki ゆうき(金野裕希)
yuuki ゆうき(金野裕希)

infinities are written
• ℵ₀, ℵ₁, ℵ₂, …, ℵ_ω, …
in order from smallest to largest.

for the number of reals ℶ₁,
• ℶ₁ = ℵ_?
is not provable from the standard axioms of mathematics.¹

the ℶ₁ = ℵ₁ conjecture is called the
• continuum hypothesis (CH).

yuuki ゆうき(金野裕希)<p>¹ already known to the Jains of India (400 BC), tho.<br />but they made mistakes such as the number of points on a line |ℝ| and a plane |ℝ²| are not equal. (both are 2^ℵ₀.)</p>
yuuki ゆうき(金野裕希)<p>actually, the number of natural, integer, and rational numbers are all the same.</p><p>that infinite number is called ℵ₀ (aleph zero) or ℶ₀ (beth zero).</p><p>Cantor (1874) proved that the number of reals is larger than that.¹<br />that is called 𝔠 = 2^ℵ₀ = ℶ₁ (beth one).</p>
yuuki ゆうき(金野裕希)<p>the idea that having the same number is the same as having a 1-1 correspondence is called<br />• Hume&#39;s principle.¹</p><p>¹ neither Hume nor Galileo thought it applied to ♾, tho (unlike Cantor).</p>
yuuki ゆうき(金野裕希)<p>have a look at this strange pic.<br />there&#39;s 2 circles.<br />a big one and a small one.</p><p>however, both circles consist of the same number of dots.<br />(there&#39;s a 1-1 correspondence.)<br /><a href="https://blog.wolframalpha.com/2010/09/10/transfinite-cardinal-arithmetic-with-wolframalpha/" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="ellipsis">blog.wolframalpha.com/2010/09/</span><span class="invisible">10/transfinite-cardinal-arithmetic-with-wolframalpha/</span></a></p>
yuuki ゆうき(金野裕希)<p>♾ is such a strange thing.<br />the whole and part can be the same size.¹<br />cf.<br />• Galileo&#39;s paradox.</p><p>¹ contrary to Euclid&#39;s 5th Common Notion (300 BC).<br />although some say that some CNs are by, e.g., Theon of Alexandria (4c).</p>
yuuki ゆうき(金野裕希)<p>mathematically, the answer is A✔<br />they&#39;re the same.<br />(even tho evens are part of integers.)</p><p>this is bc one integer corresponds to one even.<br />1 ↦ 2<br />2 ↦ 4<br />3 ↦ 6<br />4 ↦ 8<br />5 ↦ 10<br />︙<br />n ↦ 2n<br />︙</p>
yuuki ゆうき(金野裕希)<p>A) same bc they&#39;re infinite<br />B) infinity can&#39;t be compared<br />C) integers ofc bc evens are only half of them</p>
yuuki ゆうき(金野裕希)<p>let me explain from scratch.</p><p>Q: which is greater,<br />• the number of all integers<br />• the number of all even numbers</p><p>both are infinite.<br />there&#39;s many ways to think abt it.</p>
yuuki ゆうき(金野裕希)<p>mathematicians consider infinity a number.</p><p>this is bc in the 19c, a man named Cantor found that there are &quot;multiple infinities&quot;.<br />i.e., infinity 1, infinity 2, infinity 3, ….</p><p>just like 1, 2, 3, ….<br />can also do arithmetic with ♾.</p>
yuuki ゆうき(金野裕希)<p>im summarizing my tweets here rn<br /><a href="https://min.togetter.com/kb628xT" target="_blank" rel="nofollow noopener noreferrer" translate="no"><span class="invisible">https://</span><span class="">min.togetter.com/kb628xT</span><span class="invisible"></span></a></p>
yuuki ゆうき(金野裕希)<p>The largest number is a &quot;contradiction&quot;.<br />let me explain this.</p><p>u may be reminded of a googolplex or graham&#39;s number.<br />the problem is: graham&#39;s number+1 is larger.</p><p>they&#39;re still finite.</p>
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