KiloAlpha,
Take a minute and think things through here. First off, we need to know what the SPECIFIC language was for a training contract. Did you sign a paper? If so, what does that piece of paper say IN EXACT TERMS.
Did you make a verbal agreement? If so, are verbal agreements binding in the state that you live and under what conditions? (IE witnesses).
Does the agreement state that the owner can sue for lost revenues in that contract? Did you sign the contract under duress (Most people don't). What is the value of the contract? (IE is it for reimbursement of training costs? A specific dollar amount? Is it prorated?)
Also, despite what people might say about bleeding a turnip, there might be repercussions later. He might not be able to garnish your wages, or recoup money, but if you lose and the courts issue a judgement against you, then it can affect your credit rating.
Why not see if you can reach some other agreement with him... is it possible that you stay on while somebody else is hired and trained? (Perhaps you could pay for their training if it's only 7K?)
The truth is that it's to both of your benefit if you can reach a reasonable agreement. You might stress that if he holds you to the contract now, then it's much more likely that you leave without notice down the road. See if you can offer something reasonable to the airline as well- odds are if you are upfront and honest about the situation, they are willing to work with you as well.
When you call, you can say that you are very eager to be employed by them, but was wondering if it was possible to delay your start date due to obligations to your current employer. You might say that your departure puts a burden on the flight operation, and you would like to stay on until a replacement can be hired and trained (about 3 months). You don't even have to mention that you are under a training contract.
This way it sounds like you are dedicated to your employers, and while you are very excited about starting with the airline, you don't want to hang your current employer.