METRIC SPACES, NUMBERS AND THE STRAIGHT INTRODUCTION
Given a minimal set of axioms generating consistent formal system, denial of one of their axioms would result in a consistent formal system, but unlike the original. Therefore, consistency is not just what matters in a formal system, but is able to adapt to some schemes and external guidelines, preconceived. What follows is an explanation can be understood that conflicts with the theory of sets and does, as part of a finitary approach to mathematics.
Since the media is running the mathematics are necessarily finite, all operations mathematics to develop a hardware must conclude, to be considered a result. This leads to the following reasoning.
the different metrics
Values \u200b\u200b10, 12, 23 ..... are expressed in metric decimal or decimal language is preferred. The decimal metric has a certain power, which is related to the values \u200b\u200bthat can represent, which is given by the way it builds its values. It is possible to translate some values \u200b\u200bbetween different metrics by some process. A decimal metric can be translated for example the binary numbers.
But this metric is not complete, you can not translate it any size that has been defined in any metric imagined, as seen later.
Apparently another metric is the rule and compass, which builds its values \u200b\u200bthrough the various operations that can be done with a ruler and compass. The metric ruler and compass algebraic meet some requirements so that it is not necessary to work with a real ruler and compass to generate the magnitudes involved, it can be done through the operation of equations. There is a reason why only the algebraic treatment should be considered is that an algebraic relation that represents two straight lines intersect occupies a finite space in the memory of a hardware, while finding the common point between two lines in the memory by comparing the points of each of the lines not available to any memory capacity. When a geometry or a robot developed for this purpose, it operates with a ruler and compass, it keeps in mind a series of circles and straight lines, but a series of logical relationships between different figures, and any show of its kind to be held but a demonstration would not be algebraic.
New metrics can be defined. In general a system will metric when the entities that can generate can be compared in terms of greater or less, and can therefore be ranked a set of values \u200b\u200bthat have been expressed in this metric.
As noted, some values \u200b\u200bobtained in a given metric can be transferred to other metrics, such as the number 5 of the decimal metric can be on the line of the metric ruler-compass, if you previously chose a segment which has the value 1. Conversely can translate a value of the metric ruler-compass to a decimal number, for example, the point has been generated by the ruler and compass that is in the center of the segment determined as [0.1] can be translated into the decimal metric 0.5. But it is true that all reachable points of the metric ruler - measure can be translated into decimal metric, the metric ruler - measure (RC) is more powerful than the decimal. For example, the hypotenuse of a right triangle with legs 1, (root of 2) is a number that can not be represented in decimal form.
contrast all decimal numbers are translatable to the metric RC, the RC metric is the power of the decimal metric, and even more power.
THE ALGORITHMS
An algorithm is executing a predetermined set of instructions. To run an algorithm is required to clear this algorithm, an input and a memory which stores the state in which the algorithm is running as you input. Upon completion of the algorithm will give an ouput or output.
NUMBERS
Natural
magnitudes
The numbers are expressed in a database, either decimal, binary, hexadecimal ..... the words could be a metric numbers of base 28. Any number natural can be constructed by joining the unit multiple times. A natural number is the result of the implementation of an algorithm that has a beginning and an end, and throws a well-defined value. The numbers are constructed by following the instructions number generator which provides the metrics considered.
Fractional
is primarily a fractional representation of a quantity in a different metric to decimal value eg 6 / 8 contains a sign that a metric does not recognize. To translate a fractional to decimal metric which defines the natural need to run a algorithm, a transformation. The algorithms that are useful are those which maintains the order relation (an expression is higher or lower than the other) that has a fractional in metric to translate the decimal metric are also interesting algorithms that maintain a relationship of proportionality if this possible. For example an algorithm that maintains the order relation between 1 / 2, 1 / 3 and 1 / 4 is one that would make the image more than 1 / 2 followed by the image of 1 / 3. If also preserves the proportionality, the ratios between them should be retained. After running the algorithm gets its decimal equivalent.
But not all the algorithms that translate fractional conclude, for example 10 / 2 is just a finite number of steps to give 5, while 1 / 3 never stops running. The fractions that do not end run the algorithm does not have a value of algorithm, and have no equivalent in the decimal system. Some values \u200b\u200bwould be outside the domain of the decimal. If you run enough of these fractional algorithm, we obtain a value increasingly useful for the activities of physical life material and instead of saying that is being increasingly useful is said to be becoming more about its usefulness complete. But this discussion is only strict inequalities, which is as presets must be operated in math, and decimal base is unable to give a mathematical equality in these cases.
Irrational
The metric a / b is not the only algebraic basis, there are many other ways to order character and provide them with an order relation. An example are the roots, are represented as x ^ (1 / y), it is possible to define an algorithm to translate it to other metrics such as fractional or decimal. But the true measure of the roots for any algorithm that transforms the decimal base keeping the order relation, not all roots concluded for the decimal base. There are also roots conclude that even in an algorithm to translate it in relation to an order to a fractional basis, these roots are one of the irrational expressions. In general given an expression of a metric and algorithm translator fractional basis, is said to be irrational if the algorithm is not conclusive.
While the reverse can occur, ie a fractional can be translated into at least one root, so the roots have the same power and even more than fractional and natural. Within
irrational, there are classes, there are some irrational metric space belonging to the series can not be expressed in the form of roots or logs ect. I do not know if there are or can be constructed metric spaces that can not be irrational reduced to infinite series.
the order relation
In a system in which some characters are recognized, there is a procedure for generating expressions and there is the order relation between these expressions, there is a metric space. To decide the order of a few expressions of the system is necessary to define clearly the rules of order, which can be obtained by algebraic operation, do depend on how it's constructed or defined on the occasion said.
There is another way to order a protoespacio metric that lacks the order relation, we decided an algorithm to translate their expressions to another metric space where the result is ordered, so that the order relation is transferred from another metric space. In such systems deferred, the system depends on the algorithm since the space does not have its own order relation.
is important to note that confer an ordering a protoespacio delayed in some cases it is necessary that the translation algorithms in order to space completion, simply iterated enough times to ensure that one will be bigger than the other. But a priori not guaranteed that any two expressions of protoespacio was found that one of them is greater than the other at some point, the algorithm can not be completed order. Therefore can not be guaranteed a priori that an area without its own order relation is completely ordered in a delayed order relation.
THE STRAIGHT
previously commented the metric ruler and compass (RC), a hardware always have to operate in algebraic form, however the magnitudes in this metric can broadly expressed in a geometrical (distance between intersections ...). As noted, not all built in RC magnitudes can be translated into a decimal system, if it is further assumed that a number is a decimal scale, it must conclude that not all RC magnitudes can be achieved in number at a time.
Now the RC metric is not able to reach any point on a line. How can you know there is a point on the line that RC is not enough?. The magnitudes RC is capable of generating converted into algebraic variables have algebraic characteristics. If a quantity does not satisfy these characteristics algebraic translation into RC fails, these magnitudes are in fact. The translation algorithm applied to these expressions would not finish and never more useful successive values \u200b\u200b(approximate), but useless for a machine that makes strict mathematical. You could say that although it is impossible to say in RC, this magnitude must be algebraic over a line of RC, since it has the impression that it is bounded in each iteration. But this is a false conclusion, just as there is an irrational decimal in a sequence as long as you like, because you can not define any sequence that is strictly the irrational, it can not be on the line RC magnitude can not noted.
The magnitudes can be constructed depends on the metric space that contains instructions for such construction, another metric space may be able to extract more expressions of the same number of characters, but I can also speak of a power character regardless of a way to build expressions from these characters.
COMPLETE RC
Maybe if you added new tools to the ideal ruler and compass (and translating them into algebra) could be generated and reported more magnitudes on the line. You could add a machine to spirals, pins and string to make ellipses and other curves. This system would be a RC +, improved. It is possible that the transcription of some expressions that were previously not conclusive in this new RC + now conclusive result with the new tools. But no guarantee that any expression belonging to a metric space can be represented in the line for RC +, and it is possible that any improvement in RC can never represent some values \u200b\u200bin a finite number of iterations and these definitely do not belong to the line.
