# Tag Info

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

### 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
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
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

### 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
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
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 $8$.
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

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}$It means that you cannot apply the uniform distribution to the set of natural numbers, period. A computer that does cannot possibly exist. Therefore you'll a) have to be more specific about what "... 1 vote ### How do we compare the set of natural numbers in different models of ZFC Perhaps the most straightforward way of exhibiting a larger model$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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