I'm trying to create a UISlider that lets you choose from an array of numbers. Each slider position should be equidistant and the slider should snap to each position, rather than smoothly slide between them. (This is the behavior of the slider in Settings > General > Text Size, which was introduced in iOS 7.)

The numbers I want to choose from are: -3, 0, 2, 4, 7, 10, and 12.

(I'm very new to Objective-C, so a complete code example would be much more helpful than a code snippet. =)

Some of the other answers work, but this will give you the same fixed space between every position in your slider. In this example you treat the slider positions as indexes to an array which contains the actual numeric values you are interested in.

@interface MyViewController : UIViewController {
    UISlider *slider;
    NSArray *numbers;
}
@end

@implementation MyViewController
- (void)viewDidLoad {
    [super viewDidLoad];
    slider = [[UISlider alloc] initWithFrame:self.view.bounds];
    [self.view addSubview:slider];

    // These number values represent each slider position
    numbers = @[@(-3), @(0), @(2), @(4), @(7), @(10), @(12)];
    // slider values go from 0 to the number of values in your numbers array
    NSInteger numberOfSteps = ((float)[numbers count] - 1);
    slider.maximumValue = numberOfSteps;
    slider.minimumValue = 0;

    // As the slider moves it will continously call the -valueChanged: 
    slider.continuous = YES; // NO makes it call only once you let go
    [slider addTarget:self
               action:@selector(valueChanged:)
     forControlEvents:UIControlEventValueChanged];
}
- (void)valueChanged:(UISlider *)sender {
    // round the slider position to the nearest index of the numbers array
    NSUInteger index = (NSUInteger)(slider.value + 0.5);
    [slider setValue:index animated:NO];
    NSNumber *number = numbers[index]; // <-- This numeric value you want
    NSLog(@"sliderIndex: %i", (int)index);
    NSLog(@"number: %@", number);
}

Edit: Here's a version in Swift 4 that subclasses UISlider with callbacks.

class MySliderStepper: UISlider {
    private let values: [Float]
    private var lastIndex: Int? = nil
    let callback: (Float) -> Void
    
    init(frame: CGRect, values: [Float], callback: @escaping (_ newValue: Float) -> Void) {
        self.values = values
        self.callback = callback
        super.init(frame: frame)
        self.addTarget(self, action: #selector(handleValueChange(sender:)), for: .valueChanged)
        
        let steps = values.count - 1
        self.minimumValue = 0
        self.maximumValue = Float(steps)
    }
    
    required init?(coder aDecoder: NSCoder) {
        fatalError("init(coder:) has not been implemented")
    }
    
    @objc func handleValueChange(sender: UISlider) {
        let newIndex = Int(sender.value + 0.5) // round up to next index
        self.setValue(Float(newIndex), animated: false) // snap to increments
        let didChange = lastIndex == nil || newIndex != lastIndex!
        if didChange {
            lastIndex = newIndex
            let actualValue = self.values[newIndex]
            self.callback(actualValue)
        }
    }
}