Wanting to get a "better feel" for what $\omega_1$ might be, you will do well to consider Cantor's method for generating ordinals. Cantor stated three principles for generating ordinals : 1st …

Apr 18, 2020 · Thus $\omega_0$ and $\omega_0+1$, which are different ordinals, have the same cardinality. $\omega_0$ is the smallest infinite ordinal, i.e. the order type of $\mathbb {N}$.

Now we have that $\omega+1$ is the union of a countable set with a finite set; and $\omega^2$ is order isomorphic to the product $\omega\times\omega$ with the lexicographic order.

Well the inverse of $\omega$ is defined to be the element such that $\omega\cdot\omega^ {-1}=1$. Since $\omega$ satisfies $\omega^3=1$, $\omega\cdot\omega^2=1$. Surely then …

Dec 16, 2018 · No, $\omega^\omega$ is the set of those which can be represented as finite sequences. Namely, an ordinal below $\omega^\omega$ is a polynomial in $\omega$. So in …

3 Because $\omega$ is exactly defined to be the smallest infinite ordinal number. You may ask why is there a smallest ordinal? The answer is: because ordinals are defined in a way that …

Aug 15, 2017 · Here $\omega$ is taken to be the limit ordinal which is just $\mathbb {N}$. I am really confused as to why ordinals can't add/multiply just like natural numbers do, since they …

Nov 3, 2016 · Is there a formula that can tell us how many distinct prime factors a number has? We have closed form solutions for the number of factors a number has and the sum of those …

What's the definition of $\omega$? Are the following equivalent definition of $\omega$: $\omega$ is the initial ordinal of $\aleph_0$. $\omega$ is the least/first infinite ordinal. $\omega$ is the …

Jun 12, 2017 · $$\omega^2+\omega+1=0$$ This is the only rule I know about this lesson and of course its other forms, and i just can't figure out how to reach a number as answer for this, I …