It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.4K
- Chat 502
- Study Groups 21
- Petri Nets 9
- Epidemiology 4
- Leaf Modeling 2
- Review Sections 9
- MIT 2020: Programming with Categories 51
- MIT 2020: Lectures 20
- MIT 2020: Exercises 25
- Baez ACT 2019: Online Course 339
- Baez ACT 2019: Lectures 79
- Baez ACT 2019: Exercises 149
- Baez ACT 2019: Chat 50
- UCR ACT Seminar 4
- General 75
- Azimuth Code Project 110
- Statistical methods 4
- Drafts 10
- Math Syntax Demos 15
- Wiki - Latest Changes 3
- Strategy 113
- Azimuth Project 1.1K
- - Spam 1
- News and Information 148
- Azimuth Blog 149
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 718

Options

Let \( \text{Prop}^{\mathbb{N}} \) denote the set of all mathematical statements one can make about natural numbers. For example “n is prime” is an element of \( \text{Prop}^{\mathbb{N}} \) , as is "\( n=2 \)” and "\( n \ge 11 \)”.

Given \( P, Q \in \text{Prop}^{\mathbb{N}} \) , we say \( P \le Q \) if for all \( n \in \mathbb{N} \) , whenever \( P(n) \) is true, so is \( Q(n) \).

Define a monoidal unit and a monoidal product on \( \text{Prop}^{\mathbb{N}} \) that satisfy the conditions of Definition 2.2.

## Comments

The monoidal unit is the statement that n is a natural number. The monoidal product is the logical "and". For the demonstration of property a) of definition 2.1 I created an Excel spreadsheet with columns a, b, c, d, a implies c, b implies d, (a implies c) and (b implies d), a and b, c and d, (a and b) implies (c and d). Consider a, b, c, and d to be elements of \( \text{Prop}^{\mathbb{N}} \). Now construct 16 rows with all possible combinations of true and false for a, b, c and d. Calculate the remaining columns using logical functions (note for implication use OR(NOT(x), y)).

Property a) assumes (a implies c) and (b implies d), so only those rows for which this column is true should be considered. There are nine such rows. Now examine the final column. All nine rows are also true. Since this an exhaustive set of possibilities and we have a tautology so property a) is verified. Properties b) through d) are obvious.

`The monoidal unit is the statement that n is a natural number. The monoidal product is the logical "and". For the demonstration of property a) of definition 2.1 I created an Excel spreadsheet with columns a, b, c, d, a implies c, b implies d, (a implies c) and (b implies d), a and b, c and d, (a and b) implies (c and d). Consider a, b, c, and d to be elements of \\( \text{Prop}^{\mathbb{N}} \\). Now construct 16 rows with all possible combinations of true and false for a, b, c and d. Calculate the remaining columns using logical functions (note for implication use OR(NOT(x), y)). Property a) assumes (a implies c) and (b implies d), so only those rows for which this column is true should be considered. There are nine such rows. Now examine the final column. All nine rows are also true. Since this an exhaustive set of possibilities and we have a tautology so property a) is verified. Properties b) through d) are obvious.`

It would seem that I don't understand how to use this editor. I tried to copy the formatting in Fredrick's problem statement post and it didn't work. Also there is no way to back out of a comment you wish to delete. Arg!

`It would seem that I don't understand how to use this editor. I tried to copy the formatting in Fredrick's problem statement post and it didn't work. Also there is no way to back out of a comment you wish to delete. Arg!`

You need to select the gear, in the upper right corner, and then 'view source'. You can copy that. Although you cannot delete a comment you can 'edit' it, again via the 'gear'.

`You need to select the gear, in the upper right corner, and then 'view source'. You can copy that. Although you cannot delete a comment you can 'edit' it, again via the 'gear'.`

Wow, that is odd. I did exactly that and then when I rendered it using the save comment button I still got all of the formatting characters. Just now after refreshing the page it rendered properly. Ok, I guess it is just a timing issue.

Thanks for responding to my my post.

`Wow, that is odd. I did exactly that and then when I rendered it using the save comment button I still got all of the formatting characters. Just now after refreshing the page it rendered properly. Ok, I guess it is just a timing issue. Thanks for responding to my my post.`

@PeterGates Oh, It is a MathJax thing. At least part of the rendering happens locally. It takes some time to download the relevant javascript libraries. So, you will see a basic text representation followed by the properly rendered version. Once you have the libraries rendering will be snappier.

`@PeterGates Oh, It is a MathJax thing. At least part of the rendering happens locally. It takes some time to download the relevant javascript libraries. So, you will see a basic text representation followed by the properly rendered version. Once you have the libraries rendering will be snappier.`

I found two symmetric monoidal structures on \( (\textrm{Prop}^{\mathbb{N}}, \leq) \).

One is $$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) $$ and the other one is

$$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) $$ If that looks similar to monoidal structures on \( \mathbb{B} \), it's because it is. Given a natural number \( n \in \mathbb{N} \), there's a monotone map \( f_n: (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}}) \rightarrow (\mathbb{B}, \leq_{\mathbb{B}}) \) given by assigning true to \( f_A(x) \) whenever \( x \) holds for \( n \). It's strict monoidal and gives us: $$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{true}, \land) $$ and

$$ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{false}, \lor) $$ We can check the validity of conditions for monoidal monotone maps from Definition 2.38. In case of condition (a) we get:

$$ \textrm{true} = f_n(\textrm{"n is a natural number''}) $$ and $$ \textrm{false} = f_n(\textrm{"n is not a natural number''}) $$ both of which hold. It holds that it's true that \( n \) is a natural number. It also holds that it's false that \( n \) is not a natural number.

In the case of condition (b) of definition 2.38, it holds that the conjunction of truth values of statements about natural numbers is the same as the truth value of of the conjunction of statements about natural numbers. Similar argument goes for disjunction.

`I found two symmetric monoidal structures on \\( (\textrm{Prop}^{\mathbb{N}}, \leq) \\). One is \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) \] and the other one is \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) \] If that looks similar to monoidal structures on \\( \mathbb{B} \\), it's because it is. Given a natural number \\( n \in \mathbb{N} \\), there's a monotone map \\( f_n: (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}}) \rightarrow (\mathbb{B}, \leq_{\mathbb{B}}) \\) given by assigning true to \\( f_A(x) \\) whenever \\( x \\) holds for \\( n \\). It's strict monoidal and gives us: \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is a natural number"}, \land) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{true}, \land) \] and \[ (\textrm{Prop}^{\mathbb{N}}, \leq_{\textrm{Prop}^{\mathbb{N}}},\textrm{"n is not a natural number"}, \lor) \rightarrow (\mathbb{B}, \leq_\mathbb{B}, \textrm{false}, \lor) \] We can check the validity of conditions for monoidal monotone maps from Definition 2.38. In case of condition (a) we get: \[ \textrm{true} = f_n(\textrm{"n is a natural number''}) \] and \[ \textrm{false} = f_n(\textrm{"n is not a natural number''}) \] both of which hold. It holds that it's true that \\( n \\) is a natural number. It also holds that it's false that \\( n \\) is not a natural number. In the case of condition (b) of definition 2.38, it holds that the conjunction of truth values of statements about natural numbers is the same as the truth value of of the conjunction of statements about natural numbers. Similar argument goes for disjunction.`