# Mathematics

*Friday, June 24<sup>th</sup>, 2022, at 1:35 PM Alaska Time*



## <a name="contents">Contents</a>

- [Wanattams](#wanattams)
  - [Introduction](#introduction)
  - [My New Base<sub>1</sub> Mathematics](#my-new-base-one-mathematics)
  - [Naming and Ones](#naming-and-ones)
- [Incompleteness Theorem Review](#incomplateness-theorem-review)

<a name="wanattamas"></a>

## Wanattams

<a name="introduction"></a>

### Introduction

The present work in progress on mathematics involves wanattams, or units of the base<sub>1</sub> number system, which the author is utilizing for a replacement candidate for our current base<sub>2</sub> through base<sub>n</sub> numbering systems, particularly the base<sub>10</sub> number system, commonly called our decimal system.

The conversion is not merely of semantic or translational interest, and is not trivial.

New writings will be added here in the near future. Handwritten materials have been created for this effort, and several scans are provided below, on the purpose and intent of this system of wanattams.

A "wanattam" is an alternative conception of an initial unit of numbering which is related to the use of the number one.

<br /><img  class="ui huge center image transition visible" data-src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-1.jpg" src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-1.jpg"><br />

<br /><img  class="ui huge center image transition visible" data-src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-2.jpg" src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-2.jpg"><br />

<br /><img  class="ui huge center image transition visible" data-src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-3.jpg" src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-3.jpg"><br />

<br /><img  class="ui huge center image transition visible" data-src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-4.jpg" src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-4.jpg"><br />

<br /><img  class="ui huge center image transition visible" data-src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-5.jpg" src="24062022134702-22042022-abstract-and-summary-work-on-wanattamas-mattanaw-iphone-13-mini-mattanaw-source-5.jpg"><br />

<a name="my-new-base-one-mathematics"></a>

*Sunday, April 10<sup>th</sup>, 2022*,&nbsp;<time>1:04 PM Arizona Time</time>

### My New Base<sub>1</sub> Mathematics

After reflecting not long ago on what the least arbitrary numbering system might be, and considering certain options like binary, as an afterthought I came onto the idea that simple hashing and counting is the simplest of all, and apparently not arbitrary. Furthermore, it appears all numbers can be represented using such a system, and it appears that people would forget that their existing base10 numbering system is a convenient way of abbreviating ones. Developing on this idea over the last year or so in my mind, and with some scribblings, I have made some progressions worth recording.

I will cover observations on each of the following:

- Base1 system appears to indicate there is not a need for deimals or a zero, as has been assumed by our existing systems we have been taught and use.
- Base1 system appears to offer some clarity on when numbering should occur and when it shouldn't.
- Base1 system appears to provide clarity on what truth is and is not, in the related field of logic.
- A base1 system can be used for an alternative computer systems architecture. Instead of using binary, unary can be used. Base2 computer systems can be reinterpreted to be base1 systems potentially too (i.e. the zero in a memory word in computer system is not necessarily intended to be a numerical zero. Instead it is representative zero. Being a representative zero, it can be considered a gap or omission in unary.

First considering the initial point, let's think about how base one could be used to remove zeros and decimals in the simple system of money. We are very famiar with using two decimal places to indicate cents in the United States, and non-decimal numbers to mean dollars. It would appear we may have some difficulty if we did not have a decimal or a zero. However, in a unary system of money, one only needs to recognize what the smallest unit required is an utilize that unit. So if a cashier tells you that you owe $13.48, you already recognize that that amounts to 1,348 pennies. Which means a unary system could be adopted with only a single money value at the root.

The use of 1,348 pennies still commits to decimals however. This is also understood, although I'd say we've forgotten it, that 1,348 pennies is merely counts of single pennies. Most are familiar with hash marks. Hash marks are like the plain one, without embellishments; a vertical bar. For each and every penny you provide of the 1,348 pennies, you could have first counted with hashing an not with decimals. This means you've written 1,348 ones, and each one corresponds to one penny. In this numbering system, there is the annoyance of having to write so many ones. And really, you do need to write each and every one, if you don't have an alterntive one-recording system. Well, base10 is such a system. But base10 leads to confusions because it provides the illusion that certain numbers 1-10 have special properties. I.e. some think 7 has a special property, or that there is sevenness. There are cerainly pieces of knowledge relating to prime numbers that are of interest, but that is not the confusion I'm trying to indicate, and focus attention on. The main point is there is no 7 as a distinct number, that is special, apart from being a string of ones. Taken as a string of ones, its interest is diminished. Its relationship to superstition diminished. But more importantly, it is more clearly recognized that it is only a symbolic representation of a string of ones. Another way to make it more clear. Now that there are only ones in this unary system I've identified, just like with Chinese language, we could have a character for each and every number that exists. This also diminishes the supposed value of the symbols from one to 10 because all numbers could have unique symbols, or no unique symbol at all. All could just use 1.

Back to the issue of having to express large numbers like 1,348 with so many ones. This is an issue I am currently working on in my creation of a system of "oneing" that only uses ones but does so with much larger chunks of numbers than 10, and represents them according to requirements of perception. Here also there would be an expected ellucidation because now the symbols that represent numbers are more clearly and less-confusedly disentangled from the method of recording them. Now, it is more clear that the rules of recording numbers relate to limitations on perception, and on writing, and on printing with a computer, and on displaying variably under a large range of conditions. It is shown to be more arbitrary, and other pathways of numbering are offered up for consideration. Right now, there is a forgetfulness around numbers as being non-arbitrary, rather than arbitrary. Arabic numbering is an arbitrary system, and highly arbitrary, since apparently the amount of digits is related to our hands, which is not at all a good choice if one is wanting to be maximally unarbitrary. Notice a system of ones is the smallest possible system. When you count on your fingers, you still count ones. Other specis of animals would not choose 10, probably, but they would have no choice but to choose one. In this sense it is not even a speciesist system to use unary, but it is a speciesist system to use base10; or at least, it would be more inviting to other species having 10 fingers or toes like we do, but looking across the animal kingdom, it appears very many animals do not have 5 fingers and two hands. This appears to be a consideration that is not necessary. However, there are other relationships the numbering system has to non-animal considerations, like creating computers. And optimal ways of creating computers have nothing to do with fingers. But better still, mathematics is something that strives for permanence and timelessness. If permanenet many billions of years must be now accounted for in ones, and must include any animal that might use the system that might be offended if they are not numbered among the mathematical. Also, it is to be expected that animals would be victims of systems employing nummbers, and it is useful to not forget that they could be beneficiaries of the math and not only humans, in ways the reader may not recognize immediately. We should not consider that we would be the only species forever to use math, particularly because in the near future, homo sapiens would not be the species that would be using that math that we are creating. They may not even have 10 fingers and 10 toes, because likely that will be something we can choose and design for, and later it is highly likely other options will be considered and chosen. In any case, however, it appears ones are required and are non-biased. It appears that base1 is the least arbitrary number system, apart from our special needs of reading, recording, and perceiving differences. My new numbering system will address these issues as the arabic numbering system already has, but without forgetting that that is a separate consideration from the numbering of things in ones.

*[Stopping point, 1:33 pm. Total writing time 19 minutes with no edits]*


<pre><a name="create-a-gap"></a><div class="ui section divider"></div></pre>

<a name="names-and-ones"></a>

*Saturday, July 30<sup>th</sup>, 2022*,&nbsp;<time>5:23 PM Alaska Time</time>

### Naming and Ones

In the prior section a simple example case where replacement of a base<sub>n</sub> sytem with a base<sub>1</sub> system of wanattams is

- feasible
- removes the need for the decimal place
- removes the need for zeros

It was also discussed, why a system of base<sub>1</sub> is a non-speciesist, more future-resistant non-arbitrary system of numbering, because while alternative systems have in their history made commitments relating to cognitive limitations, or numbers of fingers and toes, this system has no such commitments. It was discussed that this system is a necessary system because whatever system is employed, at a minimum counting with ones is required.

It may be additionally remarked, that any non-base<sub>1</sub> system is already considered a translation equivalent to any other system of a non-one base. A base10 system, being a translation of a base1 system, can be rewritten as a base one system, which implies that all mathematics relying on base10 is simply another way of writing base1. The cause of not using base1, it was discusses, relates to the need for not writing too many digits when representing large numbers.

The present section has the following additional developments and interests:

- Wanattams or base<sub>1</sub> can clarify when names can be applied and cannot be applied.
- Rules for the application or naming of things with wanattams, determine, in conjunction with the practical/pragmatic needs for using math, what operations may be employed. In the system of Wanattaming, it is not assumed for example that division may be employed in any scenario formerly considered standard for division. 
- Avoidance of fractions is discussed, in the context of the earlier avoidance of using a decimal, or period with decimal digits, and the additional discussion, of the relationship of division to naming.

Let's develop these points with a simple case of deciding how one will divide a pizza. This is one very simple example that is intended to be illustrative of a much wider and abstract need, which will be used as a starting point for subsequent examples which will be increasingly complex. The use of the division of a pizza is uniquely interesting in that it is very straightforward and well understood in our experience, but can be used to show that there are very serious errors in our usage of mathematics, and our assumptions about what math is and how it can be used.

Suppose you order an extra large pizza at a restaurant. You have 7 total people including yourself in your party. You may order more food but initially you will have this first pizza, which you will need to cut into pieces, to serve each person a different portion.

When the pizza is served at the table, it is uncut. The waiter/server stands over the table ready to do long-division on this extra-large pizza, using long cuts from one end to the other, through the center. Before cutting, he asks you how you would like it to be cut, in order to satisfy yourself and each of your guests optimally. 

You are a mathematician and you can't simply use what you've learned in school. You develop upon it inorder to extend the math that our civilization can utilize. So you think critically about this situation and try to quickly find an optimal way to divide the pizza.

Observing the pizza closely, you notice it is not an exacting pie. It is larger on one side than the other, and it isn't totally circular. Additionally, rotating the pizza in your mind, you notice that a cross section of the pizza would indicate different densities in slices, and that some parts would be thicker than others. The other guests, wanting to assist you, are looking mostly towards one side of the pizza, which has been more favorably supplied with favorite ingredients, and more cheese. Normal application of mathematics will not be adquate in this case, you know, because it will simply call for a division of the pizza with cuts, the placement of which are ignored, that will result in an "eyeballed" separation of the pie into 8 pieces. This would be achieved with four cuts across the pie, resulting in 8 semicircles. There is no clear center point in the pie, just a point where the cuts hopefully all cross each other, or nearly do. 

It can be seen quickly, by an imaginative reader, that there are many problems in addition to the ones listed above. If one thinks about how one would make a pizza fair for children, one would find that there is no way to arrive at equality in the pizza long-division, and the kids will find many reasons to make you believe the cuts were unfair. The best you can do is decide for them, or make them feel satisfied that what they get is fair in other ways, or fair enough. Additional issues will be added shortly, but for now let's consider that our normal ideas about how we would apply math are not really that critical, and do seem to have problems.

Consider that we have chosen a cutting technique that assumes we want an even number of slices. This has assumed that not only will we divide, we will divide the pie *evenly*. I will argue at a later time that this assumption that there is really an even and an odd is not particularly clear in a system of wanattaming. For now I will comment that an even division of the pie, on social ideas, would be one that does not have any additional unused pizza remaining, or a modulus or remainder in regular math. If we were to have an additional slice, also, we are suddenly going to run into the same issue again, of allocating that slice, potentially for the seven people, who have not yet had enough food to satisfy them. So once again, division seems like it needs to be employed. But now consider, that slice is not a circle, and the former assumptions about cutting with four long-divisions into 8 slices won't work as effectively. One would also be disinclined to cut that remaining slice into 7 slices, not being used to such cuts, and also because, the cuts are so small as to not be socially acceptable. The result is that there is not a clear application of mathematics to the division of the pizza.

That additional slice also reveals that this system would rely on fractions or decimals of the pizza. Now, every person did not get one slice, but received one and one-sevenths or ~0.1428... an awkward number. This readiness to have a decimal, and a zero, and a fraction is not really justified in a way, that makes it so that another method of even divions into 7 slices would not be better. 

A system of wanattams here would call for the division of the pizza into the minimal number of slices that make sense given the social rules and the physical work to be performed. It seems here, that 7 slices is adequate, but 14 slices would work as well, if each person could get 2 slices, from different parts of the people in a way that allocates pizza resources more equally. Notice that simply cutting the pizza into 8 slices does not actually include allocation of resources in an equal way, yet the number 1 has been applied to each of the slices. The result is that the pieces are not really even the results of a division operation the preserves an equality relation, and that not one piece is really equal to the other, has not been used in any socially approved way of equally allocating resources, and instead is a reliance on school division and an easy method of cutting. A system of wanattams, however, would want to divide the pizza into 7 slices, in a way that is a proper application of math according to the needs of the situation, with the result that any 7 slices are also trully equal to each other. However, in this case we will find that we do not have determinate needs or requirements for cutting, and pizza allocation, so the result is that wanattams will still fail on the requirement that each pizza be equal, or really have a wannatam value of one applied.

There are two serious issues here with this pizza long division that the mathematician using wanattams needs to resolve. Firstly, there is the issue that the math used does not employ physics. The second is that which has already been explained, that the method of cutting must also satisfy each person eating the food on social grounds of fairness, with the result that each slice is equal on other grounds. It will be assumed, since Mattanaw is an expert in moral philosophy, that for now, the second social requirement will simply be that each person ostensibly approves of what they receive, and that a sufficient level of fairness for someone like Mattanaw is achieved (I.e. there is nobody compmlaining because they really seem close to what reasonable people would expect for social justice of allocating food). Instead, you, like Mattanaw, intend to focus your attention to the physics of the matter, in order to gain an equality amongs each piece.

Notice however, that calling each piece a piece of the pie is not quite justified. When a piece is cut, and it is a wanattam, it will have a name, and that name will be one. Once the one has been applied, it will be satisfactory to refer to each slice as a singular piece. It is possible to exit this way of thinking to call piecdes and subdivision of the pie, but once the division has ocurred, it will be necessary that each piece really is a one. Then all who speak about their pieces of the pizza will be referring to the same thing, *namely*, and *numerically*, even one-sevenths of the pie, divided along physical bases, corresponding somewhat to social expectations of allocation of resources.

So you look at this pie, and you think to yourself, how can I optimally cut this thing so it has about equal amounts of cheeze, crust, toppings, etc... with similar visually pleasing properties which can be defined in terms of colors and arrangement. You also want to weight the pizza, in order to estimate whether or not a particular cut really did arrive at what you anticipated were your requirements, meaning you will have to test the pie after it is divided to see if the division was right.

To keep conversation short, let's simply say you arrive at what you think is equal in terms of wanattam assignment and division into seven equal portions, using physical properties, and you are able to communicate the need to divide the pizza along specific lines segments. Let's say also, you look to your group, and they, showing signs of distrust in your judgement, hear you out on your plan, and vote unanimously to support your decision about how to cut. The Server/Waiter then cuts the pizza skillfully, despite having an inclination to cut with 4 strokes. 7 separate strokes cutting into the center of the pie, which was also roughly determined in a manner similar to center of gravity.

Now, everyone eats the pizza, and is very happy that none is leftover to look at, to think about how it will be divided once everyone finishes their pieces. There was no use of a fraction. There was no use of a decimal. Each piece, though definitely unequal, was physically examined to create rough conditions for equality, good enough to pretend to well implement math, using the wanattaming appraoch. The slices were evenly distributed to each guest, even though the number in base<sub>10</sub> would have been a prime odd number. In your mind, as you were going through the allocation, were thinking in ones. You did not really think there was any seven involved. You thought you were naming sections of the pizza, pieces and that each would also be named one, and that there would be

1111111 slices.

Notice also that the base<sub>10</sub> seven was clearly translatable to ones without any issue or loss of function. Notice that this different application of math without assumptions arrived at a more clear result, than any use of pizza long-divion that you or anyone you know has ever used.

We have seen each of the earlier ideas about wanattams shown functional. There was no need for a decimal, or decimal places, and no need for fractions. There was no need for a unit smaller than 1. 

We have also seen that each of the other topics of interest mentioned were covered. Firstly, that social justifications were required in order to determine which math was applicable, even for something as simple as division. Arguably, division was not performed, but naming of the pizza. What exactly ocurred was a matter of process, and not a matter of strict application of division. We've seen that use of wanattams provides clarity about naming, since before we would have individual slices that arguably should not have the same name, for being so different. Instead, having really closer equality, results in more acceptable naming, and more acceptable use of numbers, since if 1 does not equal 1, then it is clear the math has not been sufficiently well applied. In passing we saw that physics was required for this approach, connecting naming, application of ones, and physics to mathematics. We have also seen that this system clearly translates between base<sub>10</sub> and base<1>, except that using 1111111 to denote the number of slices, is more clear, because it requires less interpretation than 7, and no assumptions that 1111111 cannot be even. 

It will be found that this example provides many points that we can develop on later, and much will apply to all mathematics and physics, and linguistics/language, but also to our ideas abou what is fair and isn't.

Since this approach seems more fair and clean in its division of slices into pieces that are satisfactory, arguably, it is more fair than any division of pizza to date has been. This would imply that food allocation of pizza, unless there was luck involved (i.e. 4 people and 8 very factory-like slices), was not really that equitable, and required one or another participant to feel unsatisfied. 

Both systems would ultimately fail to provide a result that is totally considered socially acceptable or equitable, however, because like children, all would differ in how they analyze the subject according to their varying tastes and bodies. I did not consider body composition, or sex, or comfort levels about portions.

Even if wanattaming in pizza long-division cannot satisfy all such that social-justice is achieved, it does actually resolve a number of issues in the application of mathematics. And in any case, it extends our coverage as to the feasibility of a system of wanattaming or numbering with base<sub>1</sub> instead of base<sub>10</sub>.

*Written without edits in 57 minutes. Finished Saturday, July 30<sup>th</sup>, 2022*,&nbsp;<time>6:30 PM Alaska Time</time>

<pre><a name="create-a-gap"></a><div class="ui section divider"></div></pre>


<a name="incomplateness-theorem-review"></a>

## Incompleteness Theorem Review

*Started Sunday, August 21<sup>st</sup>, 2022*,&nbsp;<time>4:01 PM Alaska Time</time>

Recently I have returned to reviewing Gödel's incompleteness theorem, which is recorded in On Formally Undecidable Propositions of Principia Mathematica and Related Systems. This paper, in my estimation, is not one that is very cleanly prepared. Rather, I should say, it has many issues which contribute to its inaccessibility. Here I will record something which might clarify the work for others; however, the intention is to really clarify it for myself, so I can very clearly communicate what is true of his paper, and what the real implications might be.

I have divided my analysis of his work according to the following divisions, in the table of contents below.

### <a name="contents">Contents</a>

- [Summary of his work](#summary-of-his-work)
- [My approach to his same objective](#my-approach-to-his-same-objective)
- [Gloassary & Definitions](#gloassary-and-definitions)
- [Axioms and Justifications](#axioms-and-justifications)
- [Rules of Inference and Justifications](#rules-of-inference-and-justifications)
- [Claims of Gödel](#claims-of-godel)
- [Omissions, Sudden Introductions, Errata](#omissions-sudden-introductions-errata)
- [Dependencies](#dependencies)
- [My notes for research](#my-notes-for-research)
- [Questions and Criticisms](#questions-and-criticisms)

<a name="summary-of-his-work"></a>

### Summary of his work

In this section I restate Gödel's work in my own words.

There is a philosophical interest in Gödel's work which seems to have a zeal which encourages various thinkers to claim conclusions and impmlications which are strictly not implied by the results of his work, which is mathematical. Gödel's work, being mathematical, does apparently have intentions in the results and implications which are, really mathematical and logical. Some of these conclusions, where they really do connect with demonstrations and proofs, may have implications that extend into the philosophical. Of course, the interests of the original authors of related works are not only mathematical but philosophical too. Bertrand Russell himself is a philosopher greatly esteemed outside of mathematics. Popular works of various authors lead to thoughts about what Gödel's paper might claim, which are not really sufficiently mathematical. It is to be recalled that each writer of the papers related to the present work of Gödel was done by accomplished mathematicians, having advanced degrees specificially in math.

Here I will focus on the work of Gödel and summarize his statements and development of his argument sticking to the mathematics and logic involved, without going to far astray to what his work might mean, to a popular audience wanting to relate things to popular philosophy. It is clear that doing so would result in a lack of clarity about what Gödel was doing and what his paper really says. The objective of the present author is to understand what he says and what his paper really implies, with a focus on the mathematics in the paper and the objectives of his colleagues where those objectives are logical and mathematical.

My own interests outside of this effort are like most others: I'm interested in life and philosophy and what the conclusions might imply about things that matter inside of mathematics, but more importantly, what might be conisered outside, in everyday life and in general knowledge. 

>> "How might my general knowledge be altered by an understanding of this work, and what does it mean about my future plans?"

These considerations will be separate to those appearing in this section, but if I understand Gödel aright, withought embellishment and confusion, I will do better at answering such a question.

...


<a name="my-approach-to-his-same-objective"></a>

### My Approach to his same objective

In this section, I use my own approach, which is informed by my history in modern computer science, to accomplish the same objectives or prove the same thesis which Gödel purports to have proven. In this section, my goal is to better understand Gödel's thesis, but instead of using his proof, I use an alternative approach which is intended to arrive at a similar conclusion, or else fail to succeed, indicating that perhaps, the implications which others might draw from Gödel's work, using his work alone, are not perhaps justifiable; or are justifiable.

In my understanding of the work being considered, the objective is to determine what may or may not be proven without serious consequences or side effects within a closed system of rules, about mathematics. The objectives of various logicians creating systems like Principia Mathematica relate to seeing whether or not mathematics may be more firmly defined by more basic rules and axioms of logic, which hopes that mathematics can be given a well understood and solid foundation. In a sense, the objective is finishing the foundations of mathematics, and defining all of mathematics in terms of the simpler rules and axioms existing in the foundations.

Here it must be acknowledged that later developments in computer science seem to indicate that general purpose computing, on the basis of a few axioms and rules of inference, which are really similar to those that appear in these systems, have changed much of life on the globe, fulfilling objectives of representing and simulating parts of life, and of representing and testing mathematics used in the sciences. The sciences, relying upon math, rely also on computing systems to carry out the mathematical operations. This is all achieved on a very simple foundation of logic which really was the outcome of work from Rusell and Whitehead and others.

There are interesting questions which relate then to Gödel's work, which claims that there are serious flaws in systems, that resemble computer systems, in really representing or proving various mathematical statements, which exist outside of computing and outside of logic. There are also questions as to definability of certain mathematical statements, using computing systems. It appears in some places definition in systems like Principia Mathematica relate to provability, in that the generation of the mathematical formulae from underlying axioms and rules of inference, shoudl not be possible if the resulting formulae are not provable, using those same axioms and rules of inference which are supposed to be basic enough to be self-evident for proof. A computing system should not be able to generate or represent a mathematical statement unless that statement were already proven by the design of that computing system. 

It is the opinion of the author, knowing much about the design of computer systems in architecture and their implementations in softare, that mathematical implementations have been mostly separated from the hardware implementations which really did rely on logical design. This means that the results of mathematical operations are more related to software implementations which probably approximate mathematics rather than define them, using the rules of logic and axioms the way that the hardware appears to do in some ways, and the way that Russell and Whitehead and others have tried to do on paper in their systems of logic.

Gödel's thesis is that there are mathematical propoositions and formulae which, along with their denial, conjointly cannot be proven. In other words, the statements cannot be proven true, but also, cannot be proven false. These propositions then have an indeterminate status with respect to their correctness and truth. They are found unprovable both for their assertion and denial, and are said to be undecidable. Undecidable is taken to mean "If one claims that this proposition is true, and another claims that it is false, neither can be satisfied." It cannot be proven one way or another.

Gödel does not work towards showing that one specific example faults the approach, but rather that there would be many instances becasue of the faultiness of the approach. One might say that this is like stating that an architectural style used for building homes would result in dangerous dwellings. He supposedly shows that the entire architectural style is faulty. This is where there can be overstatement as to the implications, though. While he shows, perhaps, that an entire agenda or plan results in undecidability for a large number or infinite number of propositions, of certain kinds, it does not imply that certain errors cannot be overlooked, for being unimportant. This would be like finding if an architectural style has serious defects which lead to some undesirable appearance on a large number of occasions, but that this error is not one that does not make it less desirable overall. The implications have to be well understood in their effects for various disciplines, and what it means pragmatically for various objectives mathematicians and computer scientists have. 

It appears that Gödel's work has not blocked the progress of computing, which has its foundation in logic.

Here I try to show the meaning of Gödel's work by trying to do what Gödel did in another way, with modern knowledge about computing and systems architecture in mind. When Gödel wrote the computer did not exist at all. A field of expertise I have is computer software architecture, and this field did not at all exist at the time that Gödel was writing. Some things included in his paper are intensely interesting because they seem to presige some elements of the fundamentals of computer architecture, like how symbols would be encoded into numbers, which we know are binary. He uses a very different approach to encoding his symbols which do not include anything like the approach in modern computers, but instead he encodes everything in a sequence of natural numbers.

*To be continued...*

<a name="glossary-and-definitions"></a>

### Glossary and Definitions

Here are terms used by Gödel which require clarification and definition for readers to follow along.

- formula
- proof Schema
- proposition
- axiom
- undecidable proposition
- unprovable or provable proposition
- recursion
- class-sign
- relation
- function
- substitution
- recursively defined
- symbol
- sign
- calculable

<a name="axioms-and-justifications"></a>

### Axioms and Justifications

These are the fundamental axioms and their justifications. These are those that  Gödel is permitted to use in his proof.


<a name="rules-of-inference-and-justifications"></a>

### Rules of Inference and Justifications

These are the rules of inference and justifcations of those rules, which Gödel is permitted to use.



<a name="claims-of-godel"></a>

### Claims of Gödel

Gödel's work is one that has a number of claims that are not directly part of his thesis. These must be tracked, in addition to those which are part of his thesis, to identify what omissions may exist, and which supposed implications and conclusions are not substantiated. It is not unusual that a work have statements which were not subsequently edited out, in order to tighten up and focus the paper on the primary objective of the author. For this paper, there are many readers who do not understand, who are willing to support various purported conclusions and implications, which may or may not have realy been substantiated. For that reason I am giving the paper a careful reading to track all the claims he has really made and which of those claims have been supported, or were attempted to be supported.


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### Omissions, Sudden Introductions, Errata

The purpose of this is to track issues in his argumentation. His paper does not proceed precisely as he says it will in his introduction to the prrof, which is supposed to be less exacting. There appear to be omissions, sudden unjustified introductions of technique, and various errors in the work, which need to be tracked. These issues do not mean that Gödel has been unsuccessful in carrying out his objective, but it does imply his paper is not as easy to read and understand, and not as well edited, as one might desire. In the course of this analysis, however, it may alternatively be found that, his errors or changes and introductions of methods unexpected actually do harm to his argument, and perhaps he has not proven what he claims to have proven.


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### Dependencies

This paper utilizes many assumptions which are carried from other authors, or from the discipline of mathematics itself, in branches already developed. Reliance upon these authors, their work, already crated axioms, already created rules of inference, and other portions of mathematics may result in importation of errors. He has not developed he topic afresh and many omissions in explanation are made, with the effect that Gödel places upon the reader the burden of researching and studying all the dependencies in advance, or in the course of laborious parallel reading. There are some dependencies that do not seem justified, going on initial intuitions, which do come from experience in related areas of interest including logic, mathematics, and computer science. I will utilize this list to see which dependencies appear trustworthy and which might not be. It is unlikely given the density and length of materials Gödel relied on to definitively approve each and every dependency with authority. However, letting the reader know which portions I have taken for being reliable can be used for further analysis and potentially further corroboration or refutation of Gödel's claims.

- Peano's axioms
- Russell and Whitehead's *Principia Mathematica* and their rules of inference there adopted.
- Hilbert's logical syntax
- Zermelo-Fraenkel's work
- External mathematical techniques requiring detailed listing (later).
- Rules of Proof (proof theoretic rules relied upon).

Whatever methods of proof used by Gödel that are central to his argument require explication.

<a name="my-notes-for-research"></a>

### My Notes for Research

This section includes my notes for my own personal research. Areas where I am aware I need additional information in order to more accurately understand Gödel's statements.

For research:

- Peano's axioms.
- Russell's theory of types, and whether there is a relationship to the typelifting described by Gödel.
- A clear definition of free variables in math and logic.

<a name="summary-of-his-work"></a>

### Questions and Criticisms

In this section I list questions and criticisms, which may target Gödel's claims, but may also critique approaches of any of the other related mathematicians and their mathematics. These questions and criticisms may relate to works I have in progress that have different directions. I admit here that my inclination is contrary to Gödel's, and my work has a different trajectory. I have many questions and criticisms that either relate to his supposed results, or else the implications of his paper which may or may not be justified, given what he really shows, and what he does not show.

- How are the natural numbers that are mapped to formulae and propositions in Gödel decoded?
- How does his mapping of natural numbers to formulae and propositions compare with existing approaching in computer systems, using binary to ascii or UTF-8?

Key Criticisms:

- The work is vague in many locations and the reader is not guided on a path that would be clear to his best readers.
- Not indicating the level of analysis, when he is moving from concrete to the metamathematical, and when his statements are composed of statements at various levels which would require significant thinking to reduce or resolve.
- Lack of visual diagrams supporting the above purpose.
- Lack of definitions of certain terms relied upon, that the reader may not understand, such as his use of "type".
- Movement to methods of argument that do not appear to be planned in the beginning of the paper. In other words, one reads his supposed proof approach and sees one thing, but then gets something quite different when one reads his actual proof.
