Thanks for this interesting post, jread. I've been thinking about this topic a bit.

I think it's important that you refer to a conclusion that cannot be derived through consistent reasoning. Any conclusion that one starts with, rather than arriving at, has not been subjected to consistent reasoning (it isn't really a conclusion after all), but it isn't to say it couldn't be, and determined true or false. Are there positive claims in the wild that can't be subjected to this scrutiny?

Good question. I think this also is what magilum was essentially asking about. Looking for an example of irrationality as described in case (2) is not something that I have been able to arrive at yet. However, the foundation/explanation by which one could find an example is laid. In order to find one, I would proceed by looking at something which is unable to be derivable in terms which a consistent axiomatic system operates with. I suspect, that an example of this kind involves a deeper grasp of Goedel's theorem. By better grasp I mean:

what does an axiomatic system leave out when it sacrifices completeness for consistency?

My example of irrationality in the form of (2) would be something that is left out of the axiomatic system for the sake of consistency. I have not yet been able to grasp what something like this is.

I couldn't agree more. Although the formulation of a concept without yet having something to apply it to is not a problem for my goal because, at this point, it is a mere hypothesis. If it were a positive statement, completely lacking any sort of application definitively, then I would be in a hard place. However, being as it is still in hypothetical status, the mere fact that I can posit a hypothesis that can stand on its own (so far), yet currently lacks a concrete example, is still a pleasing thing. I do genuinely hope that my prospects for an example are not illusory. It appears to me, that because my hypothesis is based on a mathematical theorem concerning the nature of axiomatic systems, that there may be an example which can support my hypothesis which is derivable from the proofs that support Goedel's theorem. For example, that which an axiomatic system must leave out in order to be consistent. Ultimately though, I do concede that this hypothesis may not be supportable, but I am anxious to learn why until I can discover how to support it.    

I may be totally off base here, but what about self-negating statements as an example of something that an axiomatic system must leave out in order to be consistent?

This is based on Hofstadter's work with Godel's theorem and the idea that self-referential statements are problemmatic for all formal systems.  A self-negating statement such as the liar's paradox (everything I say is a lie, including this) creates incompleteness in systems of meaning, so they either have to be left logically unresolved or arbitrarily declared meaningless and ignored.

That would definitely be an intriguing point to consider, whether or not an irrational belief/thought/proposition is actually self-negating, and therefore left out of an axiomatic system because of that characteristic. Perhaps some of deludedgod's arguments concerning the very idea of God would classify beliefs of said type as unavoidably left out of an axiomatic system (I am referring to his essay on God is a broken concept).

Good point to bring up though Textom, if you have any other thoughts concerning their relation to my position please post them so we can discuss them. Like I said in my original post, this is a working hypothesis and I welcome all criticism and challenges in order to test the validity of my position. So for that, I thank you. 

Ok Vessel, I have thought about a way in which to express an irrational belief/proposition/thought in the way I have outlined.

Consider the following line of reasoning:

Concerning Belief A 

 Belief A: An eternal, omni-everything God exists.

 Is Belief A rational or Irrational?

 If rational, then it is derivable from within our consistent axiomatic system of reasoning i.e. logic.

Is Belief A derivable from within logic (a.k.a. reason)? No.

Therefore, Belief A is irrational, meaning that it is underivable from within our consistent axiomatic system of reasoning.  

Objections :

Now, it does occur to me that such a belief as 'Belief A' is not merely irrational because it is underivable from within a consistent axiomatic system of reasoning. In light of this I will try to point out what I see is problematic with this example, and then I will try to address these objections. 

Objection 1: I don't need axioms to label 'Belief A' irrational, it just is. 

My entire purpose of noting the bloated 'axiomatic system' bit is to unpack the meaning of the word irrational and the relation to what is called reason; I attempt to define reasoning in the context of the meaning of irrationality.

In this respect, I bring up axiomatic systems as an effort to perhaps capture a conception of 'irrational', where it notes the connotational ambivalence of something being irrational because it is not derivable from within a consistent axiomatic system of reasoning. 

Objection 2: Why this hype about connotations? 

In my mind, when the word irrational is attached to a proposition or belief, there are typically other meanings connoted, such as dumb, silly, stupid, idiotic,etc. Therefore, I am primarily arguing a truer conception of the word irrational in order to remove negative connotations which have nothing to do with a belief/thought/proposition/idea being irrational.  And by "truer" I mean more definitional. I am not trying to redefine the word irrational, merely re-conceive it.

Objection 3: I don't need to call Belief A irrational, I can just call it stupid, silly, or naive.

Indeed, this is always an option. However,I feel this demotes the justifiability of your statement because it turns out purely subjective. If I, me, or you call something silly, we can't prove that it's silly. We just have that emotion or belief and we firmly believe it to be true! 

In this respect, this is why I seek to remove the negative connotations attached along with the word irrational. The fact that Belief A is irrational, can be proven. Or rather, Belief A is not derivable from within our consistent axiomatic system of belief. Therefore, on this account, Belief A's irrationality is indubitable. However, Belief A's silliness and/or idiocy is contingent.   

Objection 4: You seem to be leveling all things which are irrational to a common level of how shall I say, common sensibility? What I mean is, my irrational jealousy that my wife may be cheating on me is a lot more sensible than a belief in an eternal God. Yet somehow, your conception seems to equalize the two beliefs? This is wrong. 

This objection (or any form of it) is the hardest to address. What it is suggesting, in case I was unclear, is that something irrational such as love or jealousy or hate, are things which appear/feel so real and true to the subject who is experiencing them that they almost appear rational or derivable by reason. Yet, irrational beliefs such as the one for the existence of God, seem so far away that there can be no level of comparison between an irrational love and an irrational belief in God. This I concede as a consequence of my conception. 

In order to address this, I am forced to propose a type of degree if you will, surrounding a belief/thought/proposition/idea's irrationality. In essence, a scale of irrationality would be necessary.

For example, conceive of how to gauge such a scale in terms of a metal link chain. A complete length of chain is a rational belief, having each of the links represent an aspect of that belief's derivability from within the axiomatic system. Irrational belief A would be a length of chain missing one link, therefore, eliminating the complete derivability of that belief from the axiomatic system's axioms. Now finally, consider Irrational belief B and it has five missing links. On the proposed scale, the level of incompleteness of an irrational belief has a direct correlation to the desirability of that particular irrational belief. So for example, irrational belief A only has one missing link: higher desirability. However, irrational belief B has five missing links: low desirability. Desirability in this context can be understood as referring to the other definitional aspects of  reason i.e. good judgement, sound sense.

 
Conclusion:

I hope to have to established the following:

1. Give an example of an irrational belief framed using by hypothetical conception.

2. Answer possible objections to this example.

3. Clarify the intentions behind my conception.

In summary, the importance of conceiving the word irrational and its connotations in the way I've laid out is to present a true-to-the-definition conception. In no way do I want to argue that irrational beliefs are somehow rational or should be conceived as such. Instead, my goal thus far, has been present a connotation-less conception of the word irrational in order to allow for a different understanding of the wide range of irrational beliefs/thoughts/ideas/propositions.

The purpose of this 'different understanding' is to promote the idea that things which are irrational are definitely so, yet, any other such things which are predicated of them such as silly, stupid, etc. are not so definite and in fact, themselves irrational. I want to keep the intention of the word irrational within the context of connotations that are rational. 

Note: I apologize. I realize that I introduced new ideas in my conclusion. Typically, this is avoidable. However, I feel that they serve to qualify my overall purpose. If you feel anything that I've said in my conclusion just isn't cogent or doesn't sit well, then I apologize and invite you to virtually smack me across the face for introducing new (perhaps wrong) ideas at the end of my post! 

I have changed my own paper enough to probably warrant a re-read. I cannot speak for todangst. There is also this:

All a posteriori Arguments For the Existence of God Are Intellectually Bankrupt

I thought you were going to try and justify beliefs supported by something along the lines of what Textom suggested. I have nio problem with what you have here. I have never considered irrational to necessarily carry those negative connotations. The negative aspect of irrationality is its inability to support a belief or action as a true or justified belief or action.  

This is a common opinion, but that doesn't give it any weight.  Would you care to rationally justify this?  First, what are the most important questions?  How do you know they're the most important?  In what sense are they the most important?

Have you never read about the evolutionary advantages of altruism and sacrifice?  Do you know nothing of the mathematics of game theory as it applies to cooperative situations?  Your statement reeks of ignorance.

A statement as sweeping as this must be justified, or we can dismiss it out of hand.  After all, every question we've ever answered in science or philosophy has been answered by using logic.  If there really is a magical set of questions that cannot be answered with logic, there should be a logical reason for it.