I have defined the alephs by induction : $\aleph_{\alpha+1}$ is by definition the smallest cardinal superior to $\aleph_{\alpha}.$ Is there a quick way to show that $\aleph_{\alpha+1} \cdot \aleph_{\alpha+1} = \aleph_{\alpha+1}$ assuming that $\aleph_{\alpha} \cdot \aleph_{\alpha} = \aleph_{\alpha}$ ?
2026-03-29 09:10:08.1774775408
Proving directly that $\aleph_{\alpha+1} \cdot \aleph_{\alpha+1} = \aleph_{\alpha+1}$ by recursion
81 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in SET-THEORY
- Theorems in MK would imply theorems in ZFC
- What formula proved in MK or Godel Incompleteness theorem
- Proving the schema of separation from replacement
- Understanding the Axiom of Replacement
- Ordinals and cardinals in ETCS set axiomatic
- Minimal model over forcing iteration
- How can I prove that the collection of all (class-)function from a proper class A to a class B is empty?
- max of limit cardinals smaller than a successor cardinal bigger than $\aleph_\omega$
- Canonical choice of many elements not contained in a set
- Non-standard axioms + ZF and rest of math
Related Questions in CARDINALS
- Ordinals and cardinals in ETCS set axiomatic
- max of limit cardinals smaller than a successor cardinal bigger than $\aleph_\omega$
- If $\kappa$ is a regular cardinal then $\kappa^{<\kappa} = \max\{\kappa, 2^{<\kappa}\}$
- Intuition regarding: $\kappa^{+}=|\{\kappa\leq\alpha\lt \kappa^{+}\}|$
- On finding enough rationals (countable) to fill the uncountable number of intervals between the irrationals.
- Is the set of cardinalities totally ordered?
- Show that $n+\aleph_0=\aleph_0$
- $COF(\lambda)$ is stationary in $k$, where $\lambda < k$ is regular.
- What is the cardinality of a set of all points on a line?
- Better way to define this bijection [0,1) to (0,1)
Related Questions in ORDINALS
- Ordinals and cardinals in ETCS set axiomatic
- For each cardinal number $u$, there exists a smallest ordinal number $\alpha$ such that card$\alpha$ =$u$ .
- Intuition regarding: $\kappa^{+}=|\{\kappa\leq\alpha\lt \kappa^{+}\}|$
- Set membership as a relation on a particular set
- Goodstein's sequences and theorem.
- A proof of the simple pressing down lemma, is sup $x=x?$
- $COF(\lambda)$ is stationary in $k$, where $\lambda < k$ is regular.
- Difficulty in understanding cantor normal form
- What are $L_1$ and $L_2$ in the Gödel Constructible Hierarchy
- How many subsets are produced? (a transfinite induction argument)
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?