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Premature Optimization is a Prerequisite for Success

Archive for March 2nd, 2011

From Compactness to non-standard models

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Alright, I have had enough reflection on Compactness and I want to smoothly transition to another topic. Nothing comes to mind but logic, so in this transitional post I’d like to quickly apply the Compactness Theorem to build a non-standard model of Arithmetic.

Let’s add a new constant symbol ω (omega) to Arithmetic. And let’s add an axiom schema: ω > 0, ω > 1, ω>2, … Any finite subset of this new theory has a model that interprets ω as some large enough natural number. Hence, by the Compactness Theorem, the whole thing has a model. That model is a model of Arithmetic and all arithmetic theorems are true in it. And in addition it contains a number ω that is greater than any standard integer. This is a non-standard model of Arithmetic.

Please note that the existence of this model has absolutely nothing to do with the Gödel’s incompleteness theorems or any other manifestation of incompleteness of Peano Arithmetic. I didn’t even say I was constructing my theory from Peano Arithmetic. We could take the complete True Arithmetic and achieve the same result. 

One curious thing about this new theory is that it proves all sentences of the form  ω > n, but it does not prove ∀n: ω > n. If it proved that, it would prove a contradiction, because ∀n ranges over all natural numbers defined by the theory, not just over the standard naturals. And the non-standard naturals must include S(ω) – the successor that is greater than ω. But if our theory proved a contradiction it would not have a model.

So this theory is ω-inconsistent. Per Wikipedia, “T is ω-consistent if it is not ω-inconsistent”. In the next post I’m planning to write about relationships between various consistency and soundness conditions. And Model Theory is an other melody.

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Written by bbzippo

03/02/2011 at 7:45 am

Posted in math

Disappointed by MS

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Today MS pulled the plug on Live Mesh beta – the best remote access and file sync program for PC that also once upon a time promised to become a universal sync platform of the future. I used it mostly for remote access, to connect to work from home, to home from work, and to maintain computers located halfway around the globe. Well they gave us a warning way ahead of time. They told us Mesh beta would be replaced by the new Mesh which would be part of Windows Live Essentials 2011. And the new Mesh would be incompatible with the old Mesh. The Essentials also include updates to programs that I’m using: Live Mail, Live Writer, Photo Gallery. Those are all great and useful programs and I’ve been wanting to update them, but I could not. Because the Essentials is an all-in-one pack, and you don’t get to choose which programs you want to install. You must install the new Mesh if you want to use the new Mail. And did I mention that the new Mesh cannot update or automatically uninstall the old Mesh. You must uninstall the old Mesh manually, from each PC and from each user account. That is something that is very hard to do remotely especially while being connected to the remote PC via Live Mesh! So I waited until the last day.

Today I installed Windows Live Essentials 2011 on my home PC. Did I mention that the new Essentials were pushed as an “important Windows update”. I had to uninstall the old Mesh. The Essentials installer gave me two options: “Install all Windows Essentials” or “Update the currently installed components”. Of course I wanted to update the currently installed software because the Essentials include too many programs that I don’t use. Guess what. Since I uninstalled Mesh, it was not among the “currently installed components”. So the only remaining option was to install ALL components.

I realize that Microsoft wants to serve the mass consumer and streamline the experience for the majority. But to me, the mass consumer and the majority are the users of ipads, androids, and other linuxes. I am an elite consumer – a loyal user of Windows – and I deserve treatment and experience of luxurious quality. Because, as a Windows user, I am used to luxury. Hot smile

BTW, the new Mail is good, the new Gallery is good, the new Writer is good, and Bing Toolbar sucks.

Written by bbzippo

03/02/2011 at 6:51 am

Posted in Uncategorized