MHB Proof of Second Sentence:$m<n \leftrightarrow m'<n'$

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Proof
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Wave)

I am looking at the following sentence:

For any natural numbers $m,n$ it holds:

  • $m \leq n \leftrightarrow m' \leq n'$
  • $m<n \leftrightarrow m'<n'$
  • $m<n' \leftrightarrow m \leq n$
  • $m \leq n' \leftrightarrow m \leq n \lor m=n'$

I tried to prove the second sentence like that:

$$m<n \rightarrow m \in n \rightarrow m \subset n \wedge \{m\} \subset n \rightarrow m \cup \{ m \} \subset n \rightarrow m \cup \{m \}=n \lor m \cup \{ m \} \in n$$

From the relation $m \cup \{m \} \in n$ we get that $m \cup \{ m \} \in n \cup \{ n \} \rightarrow m' \in n' \rightarrow m'<n'$.

$$m'<n' \rightarrow m' \in n' \rightarrow m \cup \{ m \} \in n \cup \{ n \} $$

$$m \in m \cup \{ m \} \rightarrow m \in n \cup \{ n \} \rightarrow m \subset n \cup \{ n \} \rightarrow m \subset n \lor m \subset \{ n \}$$

From the relation $m \subset n$ we conclude that $m \in n \lor m=n$.

How could we reject the case $m=n$ ? (Thinking)
 
Physics news on Phys.org
evinda said:
$$m'<n' \rightarrow m' \in n' \rightarrow m \cup \{ m \} \in n \cup \{ n \} $$

$$m \in m \cup \{ m \} \rightarrow m \in n \cup \{ n \} \rightarrow m \subset n \cup \{ n \} \rightarrow m \subset n \lor m \subset \{ n \}$$

From the relation $m \subset n$ we conclude that $m \in n \lor m=n$.

How could we reject the case $m=n$ ?
You could argue that if $m=n$, then $m'=n'$ and therefore $m'<n'$, i.e., $m'\in n'$, is impossible.
 
Evgeny.Makarov said:
You could argue that if $m=n$, then $m'=n'$ and therefore $m'<n'$, i.e., $m'\in n'$, is impossible.

I see! (Nod)

Could we show the fourth proposition ($m \leq n' \leftrightarrow m \leq n \lor m=n'$) like that? (Thinking)

$$m \leq n' \leftrightarrow m=n' \lor m \in n' \leftrightarrow m=n' \lor m \in n \cup \{ n \} \leftrightarrow m=n' \lor m \subset n \cup \{ n \} \\ \leftrightarrow m=n' \lor m \subset n \lor m \subset \{ n \} \leftrightarrow m=n' \lor m \leq n \lor m \subset \{ n \}$$

It holds that:

$$m \subset \{ n \} \rightarrow m=\{ n \} \lor m \in \{ n \} \rightarrow m=\{ n \} \lor m=n$$

But we know the following:

$$m \leq n \leftrightarrow m \in n \lor m=n$$

and:

$$\{ n \} \subset n$$

So, $m \subset \{ m \} \rightarrow m \leq n$.

Therefore,

$$m \leq n' \leftrightarrow m=n' \lor m \leq n.$$
 
evinda said:
Hello! (Wave)

I am looking at the following sentence:

For any natural numbers $m,n$ it holds:

  • $m \leq n \leftrightarrow m' \leq n'$

I tried to prove the second sentence like that:

$$m<n \rightarrow m \in n \rightarrow m \subset n \wedge \{m\} \subset n \rightarrow m \cup \{ m \} \subset n \rightarrow m \cup \{m \}=n \lor m \cup \{ m \} \in n$$

From the relation $m \cup \{m \} \in n$ we get that $m \cup \{ m \} \in n \cup \{ n \} \rightarrow m' \in n' \rightarrow m'<n'$.

We say that $m \leq n \leftrightarrow m \in n \lor m=n$ but then we only use the case when $m \in n$.. Do we also have to conclude something from $m=n$? :confused:
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.

Similar threads

Replies
3
Views
5K
Replies
6
Views
2K
Replies
30
Views
5K
Replies
5
Views
1K
Replies
3
Views
2K
Replies
8
Views
2K
Replies
2
Views
2K
Back
Top