hi,

i have to find a mathematical model of the following sentence, but i am stuck.

thanks for any help and advice.

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- Feb 23rd 2009, 12:29 PMsanvmathematical model
hi,

i have to find a mathematical model of the following sentence, but i am stuck.

thanks for any help and advice. - Feb 23rd 2009, 12:59 PMclic-clac
The two last parts of your formula say that the binary relation is irreflexive and transitive.

A strict order, for instance, is a binary relation which is irreflexive and transitive.

So a possible model of is:

where is the restriction to of the usual strict order on

Now, does that model verify the whole formula? (i.e. does it verify the first part?) - Feb 23rd 2009, 01:14 PMsanv
It doesn't, as it does not hold for the values 0?

- Feb 23rd 2009, 01:32 PMclic-clac
Well since the problem doesn't come from !

But indeed, that's not a model of the whole formula: there is no element strictly greater than in that set. (if is a strict order, that's what says : for any there is a such that )

So using a set whose elements are integers, a model would be... - Feb 23rd 2009, 01:55 PMsanv
Hmm, I do not get that to be honest...

Why is it:

Quote:

there is no element strictly greater than http://www.mathhelpforum.com/math-he...f2a7baf3-1.gif in that set.

- Feb 23rd 2009, 02:01 PMclic-clac
No problem :)

Since a strict order satisfies the two last parts of your formula, if we find a set and a strict order on that set which satisfy the first part of the formula, we have a model.

So assume is a strict order, the first part: means that if you take any element in your set, then you can find another element which is strictly greater. A consequence is that your set can't have a greatest element. So for instance, a simple model is

Of course, there are a lot of other models. Another one quite simple is the set of all primes with the same order. - Feb 23rd 2009, 02:01 PMPlato
- Feb 23rd 2009, 02:30 PMsanv
Thanks, I think I understand that now.

That would mean e.g. the set of all positive odd numbers will also be a possible model? - Feb 23rd 2009, 07:57 PMaliceinwonderland
The set of all positive odd numbers alone is not a model of your sentence,

You need to understand some basic concepts on what a structure (interpretation) and a model of the first-order language are.

Basically, a structure for a first-order language assigns the meaning to the parameter such that,

1. What collection of things the universal quantifier symbol ( ) refers to, and

2. What the other parameters( the predicate and function symbols) denotes.

Now, if a sentence is true in , then is a model of the sentence.

For example, consider the following sentence of the first-order language

.

(There is a natural number such that no natural number is smaller.)

The above sentence is true in the structure = with

= the set of natural numbers,

= the set of ordered pairs <m,n> such that m < n.

Similary, as clic-clac mentioned, can be the model of your sentence,