16
votes
Accepted
How do we compare the set of natural numbers in different models of ZFC
We cannot compare the $\omega$s in different models, can we? If we say that one is embeddable into another, an embedding $\omega\rightarrow\omega'$ between the two sets of natural numbers in different ...
- 237k
8
votes
How do we compare the set of natural numbers in different models of ZFC
Let $(M,\varepsilon)$ and $(M',\varepsilon')$ be models of ZFC. Remember that $M$ and $M'$ are sets (in our ambient set-theoretic universe), and $\varepsilon$ and $\varepsilon'$ are binary relations ...
- 72.3k
5
votes
Accepted
choosing a random integer from $\mathbb{N}$
Does this mean that we can’t choose a random number from all $\mathbb{N}$?
Yes, sort of. More formally, there is no way to choose a random number from $\mathbb{N}$ in such a way that every number has ...
- 409k
3
votes
Accepted
Confusion regarding the definition of $\omega$ (the set of natural numbers) in a model of ZFC
Essential point is that $\mathcal{P}(X)$ doesn’t necessarily include “every subset of $X$ you can imagine”. Rather, its members are just the subsets of $X$ that actually exist in the model. So, in a ...
- 579
2
votes
Confusion regarding the definition of $\omega$ (the set of natural numbers) in a model of ZFC
Sure, there's a 'standard cut' of $\omega$, but that's not an element of the model. Models aren't closed under (external) subsets, and $\omega$ is merely the smallest inductive set in the model.
One ...
- 55.5k
2
votes
Accepted
Is my understanding of Peano Axioms as mentioned below correct? I’ll also be grateful if questions below are answered definitively
Speaking a bit imprecisely here, but here goes…
Yes, the zero and successor function have no inherent meaning, except what is defined in the axioms.
Yes
Yes, the name of a function doesn't matter.
No,...
- 1,680
2
votes
Accepted
Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$?
Not enough reputation to comment.
Try looking at the equation modulo $8$.
Notice that $2^{3z}$ is divisible by $8$, so you would get that the sum of two fourth powers gives the residue $3$ mod $8$.
- 46
1
vote
Other ideas or help me understand the answers....BOUNTY
Put $4n+5=x^2$ so $5n+1=\dfrac{5x^2-21}{4}$ and $9n+4=\dfrac{9x^2-29}{4}$ and we need that $5x^2-21$ and $9x^2-29$ be squares. Since
$$5x^2-21=y^2\Rightarrow 5x^2\equiv y^2\pmod7$$ and $5$ is not a ...
- 28.3k
1
vote
Other ideas or help me understand the answers....BOUNTY
Assume that all three radicands are perfect squares.
The least common multiple of 4, 5, and 9 is 180 so let's do arithmetic modulo 180. Though brute force, I found that all perfect squares must be in ...
- 13.4k
1
vote
Accepted
Trouble manipulating quotients in $\mathbb{N}^\times$
We seem to be working in a context in which $\mathbb{Q}$ has not been defined, so it's forbidden to mention quotients where the denominator does not divide the numerator. The usual algebraic laws ...
- 8,487
1
vote
Accepted
Proving (rigorously) that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$
We are given a set $N$ and a set $M \subseteq N$, and we define $n = \operatorname{Num}(N)$ and $m = \operatorname{Num}(M)$. (With this setup, $m \leq n$.) We define $A_M = \{f \in B(\{1,\ldots,n\},N) ...
- 8,706
1
vote
choosing a random integer from $\mathbb{N}$
It means that you cannot apply the uniform distribution to the set of natural numbers, period. A computer that does cannot possibly exist. Therefore you'll a) have to be more specific about what "...
- 767
1
vote
How do we compare the set of natural numbers in different models of ZFC
Perhaps the most straightforward way of exhibiting a larger model $M_2$ once we have a model $M_1$ of ZFC is by the ultrapower construction. There is a natural "diagonal" embedding of the ...
- 38.2k
1
vote
Accepted
What is 'increment' in Peano Axioms?
With Axiom1 & Axiom2 , we have to take $n++$ to indicate some "Successor" Natural Number , not necessarily the "Next" Natural Number.
Here , $0++$ might be the Next Natural ...
- 6,275
1
vote
Why does Shilov exclude $0$ from natural numbers?
The author is saying that we are in general allowed to define $E+E=N$, and this results in a structure that satisfies the definition of a field. The smallest positive integer $p$ such that $p\cdot 1=0$...
- 57.4k
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