Intereting Posts

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Continuity of the function $x\mapsto d(x,A)$ on a metric space
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Intuitive definitions of the Orbit and the Stabilizer

On Page 60, Set Theory Jech(2006)

(Show that)if $\kappa$ is regular and limit, then $\kappa^{<\kappa}=2^{<\kappa}$.

It’s not difficult to show that $\kappa^{<\kappa}\geq2^{<\kappa}$. But I don’t know how to show the other way around.

- Question about Cardinality
- Proof that the set of all possible curves is of cardinality $\aleph_2$?
- Cardinal number subtraction
- The cardinality of a countable union of countable sets, without the axiom of choice
- Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.
- Cardinal arithmetic gone wrong?

I rewrite $\kappa^{<\kappa}$ as $sup_{\lambda<\kappa} \{sup_{\alpha<\kappa}{\alpha^\lambda}\}$. It looks promising, if I can switch two $sup$ operators. But I’m afraid it’s generally not legitimate which I’m not sure. I don’t find the convergence issue here as in Real Analysis.

- Infinite product of measurable spaces
- A formal name for “smallest” and “largest” partition
- Dominating strategically $\omega_1$ reals
- Correct formulation of axiom of choice
- How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?
- Prove that functions map countable sets to countable sets
- Statements equivalent to $A\subset B$
- Prove/Disprove: For any sets $X$ and $Y$, $\overline{X\cap Y} = \bar{X}\cup\bar{Y}$
- Relation between XOR and Symmetric difference
- Intersect and Union of transitive relations

Let $\lambda<\kappa$, and consider $2^\lambda$. Either $2^\lambda\le \kappa$, or not. In the latter case, $2^\lambda\le \kappa^\lambda\le (2^\lambda)^\lambda=2^\lambda$, so in fact $2^\lambda=\kappa^\lambda$.

There are now two cases: Either $2^\lambda\le\kappa$ for all $\lambda<\kappa$, or $2^\lambda>\kappa$ for all sufficiently large $\lambda<\kappa$, say $\lambda\ge\lambda_0$. In the second case, we have $\kappa^{<\kappa}=\sup_{\lambda_0\le \lambda<\kappa}\kappa^\lambda=\sup_{\lambda_0\le\lambda<\kappa}2^\lambda=2^{<\kappa}$.

In the first case, $2^{<\kappa}=\kappa$. We just need to check that then we also have $\kappa^{<\kappa}=\kappa$. But if $\rho<\kappa$, and $f:\rho\to\kappa$, then $f$ is bounded, by regularity of $\kappa$, so $f:\rho\to\tau$ for some cardinal $\tau<\kappa$. Let $\lambda=\max\{\rho,\tau\}$. Then $\tau^\rho\le\lambda^\lambda=2^\lambda\le\kappa$. We are done, because we just showed that $${}^{<\kappa}\kappa=\bigcup_{\tau<\kappa}\bigcup_{\rho<\kappa}{}^\rho\tau,$$ and the right hand side has size at most $\kappa\times\kappa\times\kappa$. Since the left hand side has size at least $\kappa$, we have that $\kappa^{<\kappa}=\kappa$, and we are done.

- Finding the fallacy in this broken proof
- For x < 5 what is the greatest value of x
- how to do such stochastic integration $dS = a S^b dt + c S dW$?
- Correspondence theorem for rings.
- Proof for gcd associative property:
- Relative Cohomology Isomorphic to Cohomology of Quotient
- Sum numbers game
- Is there a simple math behind splitting multiple bills evenly across many people?
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- Weak convergence of probability measure
- Deriving the Area of a Sector of an Ellipse
- Probability to find the sequence “Rar!” in a random (uniform) bytes stream at a position $\le n$
- Proving Abel-Dirichlet's test for convergence of improper integrals using Integration by parts
- Why aren't logarithms defined for negative $x$?
- Does a function and its Hilbert transform have the same behaviour?